Implications of X-ray Observations for Electron Acceleration and Propagation in Solar Flares

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Noname manuscript No.(will be inserted by the editor)

Implications of X-ray Observations for Electron Accelerationand Propagation in Solar Flares

G.D. Holman1, M. J. Aschwanden2, H. Aurass3,M. Battaglia4, P. C. Grigis5, E. P. Kontar6, W. Liu 1,P. Saint-Hilaire7, and V. V. Zharkova8

September 30, 2011

Abstract High-energy X-rays andγ-rays from solar flares were discovered just over fiftyyears ago. Since that time, the standard for the interpretation of spatially integrated flareX-ray spectra at energies above several tens of keV has been the collisional thick-targetmodel. After the launch of theReuven Ramaty High Energy Solar Spectroscopic Imager(RHESSI) in early 2002, X-ray spectra and images have been of sufficient quality to allow agreater focus on the energetic electrons responsible for the X-ray emission, including theirorigin and their interactions with the flare plasma and magnetic field. The result has beennew insights into the flaring process, as well as more quantitative models for both electronacceleration and propagation, and for the flare environmentwith which the electrons interact.In this article we review our current understanding of electron acceleration, energy loss, andpropagation in flares. Implications of these new results forthe collisional thick-target model,for general flare models, and for future flare studies are discussed.

Keywords Sun: flares, Sun: X-rays, gamma rays, Sun: radio radiation

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . 22 Thin- and thick-target X-ray emission . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 53 Low-energy cutoffs and the energy in non-thermal electrons . . . . . . . . . . . . . . . . . . . . . 10

3.1 Why do we need to determine the low-energy cutoff of non-thermal electron distributions? . 103.2 Why is the low-energy cutoff difficult to determine? . . . .. . . . . . . . . . . . . . . . . . 113.3 What is the shape of the low-energy cutoff, and how does itimpact the photon spectrum and

Pnth? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1 NASA Goddard Space Flight Center, Code 671, Greenbelt, MD 20771, U.S.A. E-mail: [email protected] Lockheed Martin Advanced Technology Center, Solar and Astrophysics Laboratory, Organization ADBS,Building 252, 3251 Hanover Street, Palo Alto, CA 94304, U.S.A.3 Astrophysikalisches Institut Potsdam4 Department of Physics & Astronomy, University of Glasgow, Glasgow, G12 8QQ, Scotland, UK5 Harvard-Smithsonian Center for Astrophysics P-148, 60 Garden St., Cambridge, MA 02138, U.S.A.6 Department of Physics and Astronomy, University of Glasgow, Kelvin Building, Glasgow, G12 8QQ, U.K.7 Space Sciences Lab, UC Berkeley, CA, U.S.A.8 Department of Computing and Mathematics, University of Bradford, Bradford, BD7 1DP, UK

http://arxiv.org/abs/1109.6496v1

2 Holman et al.

3.4 Important caveats . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . 143.5 Determinations ofEc and electron energy content from flare data . . . . . . . . . . . . . . .15

4 Nonuniform ionization in the thick-target region . . . . . . .. . . . . . . . . . . . . . . . . . . . 184.1 Electron energy losses and X-ray emission in a nonuniformly ionized plasma . . . . . . . . 184.2 Application to flare X-ray spectra . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . 19

5 Return current losses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . 215.1 The return current electric field . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . 225.2 Impact on hard X-ray spectra . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . 235.3 Observational evidence for the presence of the return current . . . . . . . . . . . . . . . . . 24

6 Beam-plasma and current instabilities . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . 257 Height dependence and size of X-ray sources with energy andtime . . . . . . . . . . . . . . . . . 27

7.1 Footpoint Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . 277.2 Loop Sources and their Evolution . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 28

8 Hard X-ray timing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . 328.1 Time-of-Flight Delays . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . 338.2 Trapping Delays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . 338.3 Thermal Delays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . 338.4 Multi-Thermal Delay Modeling withRHESSI . . . . . . . . . . . . . . . . . . . . . . . . . 35

9 Hard X-ray spectral evolution in flares . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 379.1 Observations of spectral evolution . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . 379.2 Interpretation of spectral evolution . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . 40

10 The connection between footpoint and coronal hard X-ray sources . . . . . . . . . . . . . . . . . 4410.1 RHESSIimaging spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 4410.2 Relation between coronal and footpoint sources . . . . . .. . . . . . . . . . . . . . . . . . 45

10.2.1 Observed difference between coronal and footpoint spectral indices . . . . . . . . . 4610.2.2 Differences between footpoints . . . . . . . . . . . . . . . . .. . . . . . . . . . . 46

10.3 Spectral evolution in coronal sources . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . 4710.4 Interpretation of the connection between footpoints and the coronal source . . . . . . . . . . 47

11 Identification of electron acceleration sites from radioobservations . . . . . . . . . . . . . . . . . 5012 Discussion and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . 52

12.1 Implications of X-ray observations for the collisional thick-target model . . . . . . . . . . . 5212.2 Implications of X-ray observations for electron acceleration mechanisms and flare models . 5412.3 Implications of current results for future flare studies in hard X-rays . . . . . . . . . . . . . 55

1 Introduction

A primary characteristic of solar flares is the accelerationof electrons to high, suprathermalenergies. These electrons are observed directly in interplanetary space, and indirectly atthe Sun through the X-ray,γ-ray, and radio emissions they emit (Hudson & Ryan 1995).Understanding how these electrons are produced and how theyevolve is fundamental toobtaining an understanding of energy release in flares. Therefore, one of the principal goalsof solar flare research is to determine when, where, and how these electrons are acceleratedto suprathermal energies, and what happens to them after they are accelerated to these highenergies.

A major challenge to obtaining an understanding of electronacceleration in flares isthat the location where they are accelerated is not necessarily where they are most easilyobserved. The flare-accelerated electrons that escape the Sun are not directly observed untilthey reach the instruments in space capable of detecting them, usually located at the distanceof the Earth from the Sun. The properties of these electrons are easily modified during theirlong journey from the flaring region to the detecting instruments (e.g., Agueda et al. 2009).Distinguishing flare-accelerated electrons from electrons accelerated in interplanetary shockwaves is also difficult (Kahler 2007).

The electrons that are observed at the Sun through their X-ray or γ-ray emissions radiatemost intensely where the density of the ambient plasma is highest (see Section 2). Therefore,

Electron Acceleration and Propagation 3

the radiation from electrons in and near the acceleration region may not be intense enough tobe observable. Although these radiating electrons are muchcloser to the acceleration regionthan those detected in interplanetary space, their properties can still be significantly modifiedas they propagate to the denser regions where they are observed. The radio emission fromthe accelerated electrons also depends on the plasma environment, especially the magneticfield strength for the gyrosynchrotron radiation observed from flares (Bastian et al. 1998).Therefore, determining when, where, and how the electrons were accelerated requires asubstantial amount of deductive reasoning.

Here we focus primarily on the X-ray emission from the accelerated electrons. Interplan-etary electrons and low-energy emissions are addressed by Fletcher et al. (2011), while theγ-ray emission is addressed by Vilmer et al. (2011), and the radio by White et al. (2011). TheX-rays are predominantlyelectron-ion bremsstrahlung(free-free radiation), emitted whenthe accelerated electrons scatter off ions in the ambient thermal plasma. Issues related to theemission mechanism and deducing the properties of the emitting electrons from the X-rayobservations are primarily addressed by Kontar et al. (2011). Here we address the interpre-tation of the X-ray observations in terms of flare models, andconsider the implications ofthe observations for the acceleration process, energy release in flares, and electron propaga-tion. Specific models for particle acceleration and energy release in flares are addressed byZharkova et al. (2011).

The accelerated electrons interact with both ambient electrons and ions, but lose most oftheir energy throughelectron-electron Coulomb collisions. Consequently, the brightest X-ray sources are associated with high collisional energy losses. These losses in turn changethe energy distribution of the radiating electrons. When the accelerated electrons lose theirsuprathermal energy to the ambient plasma as they radiate, the source region is called athick target. Electrons streaming downward into the higher densities inlower regions of thesolar atmosphere, or trapped long enough in lower density regions, will emit thick-targetX-rays. Hence, thick-target models are important to understanding the origin and evolutionof accelerated electrons in flares. Thick-target X-ray emission is addressed in Section 2.

The total energy carried by accelerated electrons is important to assessing accelerationmodels, especially considering that these electrons carrya significant fraction of the totalenergy released in flares. Also, the energy carried by electrons that escape the accelerationregion is deposited elsewhere, primarily to heat the plasmain the thick-target source regions.The X-ray flux from flares falls off rapidly with increasing photon energy, indicating that thenumber of radiating electrons decreases rapidly with increasing electron energy. Therefore,the energy carried by the accelerated electrons is sensitive to the value of the low-energycutoff to the electron distribution. The determination of this low-energy cutoff and the totalenergy in the accelerated electrons is addressed in Section3.

In the standard thick-target model, the target plasma is assumed to be fully ionized.If the target ionization is not uniform, so that the accelerated electrons stream down tocooler plasma that is partially ionized or un-ionized, the X-ray spectrum is modified. This isaddressed in Section 4.

Observations of the radiation from hot flare plasma have shown this plasma to primarilybe confined to magnetic loops or arcades of magnetic loops (cf. Aschwanden 2004). Theobservations also indicate that the heating of this plasma and particle acceleration initiallyoccur in the corona above these hot loops (see Section 12.2 and Fletcher et al. 2011). Whenthe density structure in these loops is typical of active region loops, or at least not highly en-hanced above those densities, the highest intensity, thick-target X-ray emission will be fromthe footpoints of the loops, as is most often observed to be the case. If accelerated elec-trons alone, unaccompanied by neutralizing ions, stream down the legs of the loop from the

4 Holman et al.

acceleration region to the footpoints, they will drive a co-spatial return current in the ambi-ent plasma to neutralize the high current associated with the downward-streaming electrons.We refer to this primarily downward-streaming distribution of energetic (suprathermal) elec-trons as an electron beam. The electric field associated withthe return current will decelerateelectrons in the beam, which can in turn modify the X-ray spectrum from the acceleratedelectrons. This is addressed in Section 5.

Both the primary beam of accelerated electrons and the return current can become unsta-ble and drive the growth of waves in the ambient plasma. Thesewaves can, in turn, interactwith the electron beam and return current, altering the energy and angular distributions ofthe energetic electrons. These plasma instabilities are discussed in Section 6.

The collisional energy loss rate is greater for lower energyelectrons. Therefore, forsuprathermal electrons streaming downward to the footpoints of a loop, the footpoint X-raysources observed at lower energies should be at a higher altitude than footpoint sources ob-served at higher X-ray energies. The height dispersion of these sources provides informationabout the height distribution of the plasma density in the footpoints. The spatial resolutionof the Reuven Ramaty High Energy Solar Spectroscopic Imager(RHESSI– see Lin et al.2002) has made such a study possible. This is described in Section 7.RHESSIhas observedX-ray sources that move downward from the loop top and then upward from the footpointsduring some flares. This source evolution in time is also discussed in Section 7.

If electrons of all energies are simultaneously injected, the footpoint X-ray emissionfrom the slower, lower energy electrons should appear afterthat from faster, higher energyelectrons. The length of this time delay provides an important test for the height of the accel-eration region. Longer time delays can result from magnetictrapping of the electrons. Theevolution of the thermal plasma in flares can also exhibit time delays associated with the bal-ance between heating and cooling processes. These various time delays and the informationthey provide are addressed in Section 8.

An important diagnostic of electron acceleration and propagation in flares is the timeevolution of the X-ray spectrum during flares. In most flares,the X-ray spectrum becomesharder (flatter, smaller spectral index) and then softer (steeper, larger spectral index) as theX-ray flux evolves from low to high intensity and then back to low intensity. There arenotable exceptions to this pattern, however. Spectral evolution is addressed in Section 9.

One of the most important results from theYohkohmission is the discovery in someflares of a hard (high energy) X-ray source above the top of of the thermal (low energy)X-ray loops. This, together with theYohkohobservations of cusps at the top of flare X-ray loops, provided strong evidence that energy release occurs in the corona above the hotX-ray loops (for some flares, at least). Although several models have been proposed, theorigin of these “above-the-looptop” hard X-ray sources is not well understood. We need todetermine how their properties and evolution compare to themore intense footpoint hardX-ray sources. These issues are addressed in Section 10.

Radio observations provide another view of accelerated electrons and related flare phe-nomena. Although radio emission and its relationship to flare X-rays are primarily addressedin White et al. (2011), some intriguing radio observations that bear upon electron accelera-tion in flares are presented in Section 11.

RHESSIobservations of flare X-ray emission have led to substantialprogress, but manyquestions remain unanswered. Part of the progress is that the questions are different fromthose that were asked less than a decade ago. The primary context for interpreting the X-rayemission from suprathermal electrons is still the thick-target model, but the ultimate goal isto understand how the electrons are accelerated. In Section12 we summarize and discuss theimplications of the X-ray observations for the thick-target model and electron acceleration


mechanisms, and highlight some of the questions that remainto be answered. Implicationsof these questions for future flare studies are discussed.

2 Thin- and thick-target X-ray emission

As was summarized in Section 1, the electron-ion bremsstrahlung X-rays from a beam ofaccelerated electrons will be most intense where the density of target ions is highest, aswell as where the flux of accelerated electrons is high. The local emission (emissivity) atposition r of photons of energyε by electrons of energyE is given by the plasma iondensity,n(r) ions cm−3, times the electron-beam flux density distribution,F(E, r) electronscm−2 s−1 keV−1, times the differential electron-ion bremsstrahlung cross-section,Q(ε ,E)cm2 keV−1. For simplicity, we do not consider here the angular distribution of the beamelectrons or of the emitted photons, topics addressed in Kontar et al. (2011).

The emissivity of the radiation at energyε from all the electrons in the beam is obtainedby integrating over all contributing electron energies, which is all electron energies abovethe photon energy. The photon flux emitted per unit energy is obtained by integrating overthe emitting source volume (V) or, for an imaged source, along the line of sight throughthe source region. Finally, assuming isotropic emission, the observed spatially integratedflux density of photons of energyε at the X-ray detector,I(ε) photons cm−2 s−1 keV−1, issimply the flux divided by the geometrical dilution factor 4πR2, whereR is the distance tothe X-ray detector. Thus,

I(ε) =1

4πR2

∫

V

∫ ∞

εn(r)F(E, r)Q(ε ,E)dEdV. (2.1)

We refer toI(ε) as the X-ray flux spectrum, or simply the X-ray spectrum. The spectrumobtained directly from an X-ray detector is generally a spectrum of counts versus energyloss in the detector, which must be converted to an X-ray flux spectrum by correcting for thedetector response as a function of photon energy (see, for example, Smith et al. 2002).

Besides increasing the X-ray emission, a high plasma density also means increasedCoulomb energy losses for the beam electrons. In a plasma, the bremsstrahlung losses aresmall compared to the collisional losses to the plasma electrons. For a fully ionized plasmaand beam electron speeds much greater than the mean speed of the thermal electrons, the(nonrelativistic) energy loss rate is

dE/dt =−(K/E)ne(r)v(E), (2.2)

whereK = 2πe4Λee, Λee is the Coulomb logarithm for electron-electron collisions, e is theelectron charge,ne(r) is the plasma electron number density, and v(E) is the speed of theelectron (see Brown 1971; Emslie 1978). The coefficientK is usually taken to be constant,althoughΛee depends weakly on the electron energy and plasma density or magnetic fieldstrength, typically falling in the range 20 – 30 for X-ray-emitting electrons. Taking a valueof 23 for Λee, the energy loss rate in keV s−1 or erg s−1 with E in keV is numericallydetermined by

K = 3.00×10−18(

Λee

23

)

keVcm2 = 4.80×10−27(

Λee

23

)

ergcm2. (2.3)

Here and in equations to follow, notation such as(Λee/23) is used to show the scaling ofa computed constant (here,K) with an independent variable and the numerical value taken

6 Holman et al.

for the independent variable (Λee= 23 in Equation 2.3). If the plasma is not fully ionized,K also depends on the ionization state (see Section 4.1).

Noting that vdt=dz, Equation 2.2 can be simplified todE/dNe=−K/E, wheredNe(z)=ne(z)dz andNe(z) (cm−2) is the plasma electroncolumn density. (Here we treat this as aone-dimensional system and do not distinguish between the total electron velocity and thevelocity component parallel to the magnetic field.) Hence, the evolution of an electron’senergy with column density is simply

E2 = E20 −2KNe, (2.4)

whereE0 is the initial energy of the electron where it is injected into the target region.For example, a 1 keV electron loses all of its energy over a column density of 1/(2K) =1.7×1017 cm−2 (for Λee= 23). A 25 keV electron loses 20% of its energy over a columndensity of 3.8×1019 cm−2.

If energy losses are not significant within a spatially unresolved X-ray source region,the emission is calledthin-target. If, on the other hand, the non-thermal electrons lose alltheir suprathermal energy within the spatially unresolvedsource during the observationalintegration time, the emission is calledthick-target. We call a model that assumes theseenergy losses are from Coulomb collisions (equations 2.2 & 2.3) acollisional thick-targetmodel. Collisional thick-target models have been applied to flarex-ray/γ-ray emission sincethe discovery of this emission in 1958 (Peterson & Winckler 1958, 1959).

The maximum information that can be obtained about the accelerated electrons from anX-ray spectrum alone is contained in themean electron flux distribution, the plasma-density-weighted, target-averaged electron flux density distribution (Brown et al. 2003; Kontar et al.2011). The mean electron flux distribution is defined as

F(E) =1

nV

∫

Vn(r)F(E, r)dV electrons cm−2 s−1 keV−1, (2.5)

wheren andV are the mean plasma density and volume of the emitting region. As can beseen from Equation 2.1, the product ¯nVF can, in principle, be deduced with only a knowl-edge of the bremsstrahlung cross-section,Q(ε ,E). Additional information is required to de-termine if the X-ray emission is thin-target, thick-target, or something in between. The fluxdistribution of the emitting electrons and the mean electron flux distribution are equivalentfor a homogeneous, thin-target source region.

Equation 2.1 gives the observed X-ray flux in terms of the accelerated electron fluxdensity distribution throughout the source. However, we are interested in the electron dis-tribution injected into the source region,F0(E0, r0), since that is the distribution producedby the unknown acceleration mechanism, including any modifications during propagationto the source region. To obtain this, we need to know how to relateF(E, r) at all locationswithin the source region toF0(E0, r0). Since we are interested in the X-rays from a spa-tially integrated, thick-target source region, the most direct approach is to first compute thebremsstrahlung photon yield from a single electron of energy E0, ν(ε ,E0) photons keV−1

per electron (Brown 1971). As long as the observational integration time is longer than thetime required for the electrons to radiate all photons of energy ε (i.e., longer than the timerequired for energy losses to reduce all electron energies to less thanε), the thick-targetX-ray spectrum is then given by

Ithick(ε) =1

4πR2

∫ ∞

εF0(E0)ν(ε ,E0)dE0, (2.6)


whereF0(E0) is the electron beam flux distribution (electrons s−1 keV−1). F0(E0) is theintegral ofF0(r0,E0) over the area at the injection site through which the electrons streaminto the thick-target region.

The rate at which an electron of energyE radiates bremsstrahlung photons of energyε is n(r)v(E)Q(ε ,E). The photon yield is obtained by integrating this over time.Since theelectrons are losing energy at the ratedE/dt, the time integration can be replaced by anintegration over energy from the initial electron energyE0 to the lowest energy capable ofradiating a photon of energyε :

ν(ε ,E0) =∫ ε

E0

n(r)v(E)Q(ε ,E)dEdE/dt

. (2.7)

Using Equation 2.2 fordE/dt, Equation 2.6 becomes

Ithick(ε) =1

4πR2

1

ZK

∫ ∞

E0=εF0(E0)

∫ E0

E=εE Q(ε ,E)dE dE0. (2.8)

We have used the relationshipne = Zn, whereZ ≃ 1.1 is the ion-species-number-density-weighted (or, equivalently, relative-ion-abundance-weighted) mean atomic number of thetarget plasma. Thus, the thick-target X-ray flux spectrum does not depend on the plasmadensity. However, the plasma must be dense enough for the emission to be thick-target,i.e., dense enough for all the electrons to be thermalized inthe observation time interval.Integration of Equation 2.2 shows that this typically implies a plasma density∼>1011−1012

cm−3 for an observational integration time of 1 s (see Sections 10.4 and 12.1 for more aboutthis). This condition is well satisfied at loop footpoints.

Observed non-thermal X-ray spectra from solar flares can usually be well fitted with amodel photon spectrum that is either a single or a double power-law. For a single power-lawelectron flux distribution of the formF (E) = AE−δ , the photon spectrum is also well ap-proximated by the power-law formI(ε) = I0ε−γ . The relationship between the electron andphoton spectral indicesδ andγ can most easily be obtained from equations 2.1 and 2.8 us-ing the Kramers approximation to the nonrelativistic Bethe-Heitler (NRBH) bremsstrahlungcross section (see Koch & Motz 1959 for bremsstrahlung cross-sections). The NRBHcross-section is given by:

QNRBH(ε ,E) =Z2Q0

εEln

(

1+√

1− ε/E

1−√

1− ε/E

)

cm2 keV−1, (2.9)

whereQ0 = 7.90×10−25 cm2 keV andZ2 ≃ 1.4 is the ion-species-number-density-weightedmean square atomic number of the target plasma. The Kramers approximation to this cross-section is Equation 2.9 without the logarithmic term. The bremsstrahlung cross-section iszero forε >E, since an electron cannot radiate a photon that is more energetic than the elec-tron. Analytic expressions for the photon flux from both a uniform thin-target source and athick-target source can be obtained with the Kramers and theNRBH cross-sections when theelectron flux distribution has the single-power-law form (Brown 1971; Tandberg-Hanssen & Emslie1988). The thin-target result also generalizes to the photon flux from a single-power-lawmean electron flux distribution.

For a uniform thin-target source andF (E) = AE−δ ,

Ithin(ε) = 3.93×10−52(

1 AUR

)2(

Z2

1.4

)

NAβtn(δ )ε−(δ+1), (2.10)

8 Holman et al.

giving γthin = δ +1. The photon energyε is in keV, the distance from the source to the X-raydetector is taken to be one Astronomical Unit (1 AU), a typical value of 1.4 is taken forZ2,andN is the ion column density. The power-law-index-dependent coefficient for the NRBHcross-section is

βtn(δ ) =B(δ ,1/2)

δ, (2.11)

whereB(x,y) is the standard Beta function. In the Kramers approximation, βtn(δ ) = 1/δ .Typical values for the ion column density andA, the differential electron flux at 1 keV, areN = 1018–1020 cm−2 andA= 1034–1038 electrons s−1 keV−1.

For a thick-target source region,

Ithick(ε) = 1.17×10−34(

1 AUR

)2(

Z2/Z1.25

)

(

23Λee

)

Aβtk(δ )ε−(δ−1), (2.12)

giving γthick= δ −1. The power-law-index-dependent coefficient for the NRBH cross-sectionis

βtk(δ ) =B(δ −2,1/2)(δ −1)(δ −2)

. (2.13)

In the Kramers approximation,βtk(δ ) = 1/[(δ −1)(δ −2)].

βtk(δ)

βtn(δ)

δ

Fig. 2.1 The power-law-index-dependent termsβtn(δ ) (equation 2.11) andβtk(δ ) (equation 2.13) in theanalytic expressions for the thin-target and the thick-target photon flux from a power-law electron flux dis-tribution (equations 2.10 and 2.12). Thesolid curves are for the nonrelativistic Bethe-Heitler bremsstrahlungcross-section and thedottedcurves are for the Kramers cross-section. Note that theβ axis is linear for thethin-target coefficient, and logarithmic for the thick-target coefficient.

The coefficientsβtn(δ ) andβtk(δ ) for both the NRBH and the Kramers cross-sectionsare plotted as a function ofδ in Figure 2.1. The Kramers and NRBH results are equal for


thin-target emission whenδ ≃ 3.4, and for thick-target emission whenδ ≃ 5.4. For theplotted range ofδ , the Kramers approximation can differ from the NRBH result by over90%. Forδ in the range 3–10, the Kramers result can differ from the NRBHresult by asmuch as 76% and 57% for thin- and thick-target emission, respectively.

It is important to recognize that the above power-law relationships are only valid if theelectronflux density distribution, F(r ,E) electrons cm−2 s−1 keV−1, or the electronflux dis-tribution, F (E) electrons s−1 keV−1, is assumed to have a power-law energy dependence.It is sometimes convenient to work with the electrondensity distribution, f (r ,E) (electronscm−3 keV−1), rather than the flux density distribution, especially when considering thin-target emission alone or comparing X-ray spectra with radiospectra. The flux density anddensity distributions are related throughF(r ,E) = f (r ,E)v(E). If the electron density dis-tribution rather than the flux or flux density distribution isassumed to have a power-lawindex δ ′, so that f (r ,E) ∝ E−δ ′

, the relationships between this power-law index and thephoton spectral index becomeγthin = δ ′+0.5 andγthick = δ ′−1.5.

The simple power-law relationships arenot validif there is a break or a cutoff in the elec-tron distribution at an energy less than∼2 orders of magnitude above the photon energiesof interest. Since all electrons with energies above a givenphoton energyε contribute to thebremsstrahlung at that photon energy, for the power-law relationships to be valid the breakenergy must be high enough that the deficit (or excess) of electrons above the break energydoes not significantly affect the photon flux at energyε . The power-law relationship is typ-ically not accurate until photon energies one to two orders of magnitude below the breakenergy, depending on the steepness of the power-law electron distribution (see Figures 9 &10 of Holman 2003). Thus, for example, these relationships are not correct for the lowerpower-law index of a double power-law fit to a photon spectrumat photon energies withinabout an order of magnitude below the break energy in the double power-law electron dis-tribution. Equation 2.1 or 2.8 can be used to numerically compute the X-ray spectrum froman arbitrary flux distribution in electron energy.

When electrons with kinetic energies approaching or exceeding 511 keV significantlycontribute to the radiation, the relativistic Bethe-Heitler bremsstrahlung cross-section (Equa-tion 3BN of Koch & Motz 1959) or a close approximation (Haug 1997) must be used.Haug (1997) has shown that the maximum error in the NRBH cross-section relative tothe relativistic Bethe-Heitler cross section becomes greater than 10% at electron energiesof 30 keV and above. Numerical computations using the relativistic Bethe-Heitler cross-section have been incorporated into theRHESSIspectral analysis software (OSPEX) forboth thin- and thick-target emission from, in the most general case, a broken-power-law elec-tron flux distribution with both low- and high-energy cutoffs (the functions labeled “thin”and “thick” using the IDL programsbrm_bremspec.pro and brm_bremthick.pro – seeHolman 2003). Faster versions of these programs are now available in OSPEX and are cur-rently labeled “thin2” and “thick2” and use the IDL programsbrm2_thintarget.pro andbrm2_thicktarget.pro.

The analytic results based on the NRBH cross-section have been generalized to a broken-power-law electron flux distribution with cutoffs by Brown et al. (2008). They find a max-imum error of 35% relative to results obtained with the relativistic Bethe-Heitler cross-section for the range of parameters they consider. These results provide the fastest methodfor obtaining thin- and thick-target fits to X-ray spectra intheRHESSIspectral analysis soft-ware, where they are labeled “photonthin” and “photonthick” and use the IDL programsf_photon_thin.pro andf_photon_thick.pro.

10 Holman et al.

3 Low-energy cutoffs and the energy in non-thermal electrons

One of the most important aspects of the distribution of accelerated electrons is the low-energy cutoff. The acceleration of charged particles out ofthe thermal plasma typicallyinvolves a competition between the collisions that keeps the particles thermalized and theacceleration mechanism. The particles are accelerated outof the tail of the thermal distribu-tion, down to the lowest particle energy for which the acceleration mechanism can overcomethe collisional force. Thus, the value of the low-energy cutoff can provide information aboutthe force of the acceleration mechanism. More generally, asdiscussed below, the electrondistribution must have a low-energy cutoff (1) so that the number and energy flux of elec-trons is finite and reasonable, and (2) because electrons with energies that are not well abovethe thermal energy of the plasma through which they propagate will be rapidly thermalized.Knowledge of the low-energy cutoff and its evolution duringa flare is critical to determin-ing the energy flux and energy in non-thermal electrons and, ultimately, the efficiency of theacceleration process.

3.1 Why do we need to determine the low-energy cutoff of non-thermal electrondistributions?

An important feature of the basic thick-target model is thatthe photon spectrumI(ε) is di-rectly determined by the injected electron flux distribution F0(E0). As can be seen fromEquation 2.8, no additional parameters such as source density or volume need to be deter-mined. Consequently, by integrating over all electron energies, we can also determine thetotal flux of non-thermal electrons,Nnth electrons s−1, the power in non-thermal electrons,Pnth erg s−1, and, integrating over time, the total number of, and energyin, non-thermalelectrons.

The total non-thermal electron number flux and power are computed as follows:

Nnth =

∫ +∞

Ec

F0(E0)dE0 =A

δ −1Ec

−δ+1 electrons s−1 (3.1)

Pnth = κE

∫ +∞

Ec

E0 ·F0(E0)dE0 =κEAδ −2

Ec−δ+2 erg s−1 (3.2)

The last expression in each equation is the result for a power-law electron flux distributionof the formF0(E0) = A ·E−δ

0 . The constantκE = 1.60×10−9 is the conversion from keVto erg.Ec is a low-energy cutoff to the electron flux distribution. These expressions arevalid and finite forδ > 2 andEc > 0. We call this form of low-energy cutoff asharp low-energy cutoff. An electron distribution that continues below a transition energyEc that has apositive slope, is flat, or in general has a spectral indexδlow < 1 also provides finite electronand energy fluxes, but these fluxes are somewhat higher than those associated with the sharplow-energy cutoff.

For this single-power-law electron flux distribution with asharp low-energy cutoff, thenon-thermalpower(erg s−1), and ultimately the non-thermalenergy(erg), from the power-law electron flux distribution depends on only three parameters:δ , A, andEc. Observationsindicate thatδ is greater than 2 (Dennis 1985; Lin & Schwartz 1987; Winglee et al. 1991;Holman et al. 2003). Hence, wereEc = 0, the integral would yield an infinite value, a decid-edly unphysical result! Therefore, the power-law electrondistribution cannot extend all theway to zero energy with the same or steeper slope, and some form of low-energy cutoffin


the accelerated electron spectrum must be present. As we will see, the determination of theenergy at which this cutoff occurs is not a straightforward process, but it is the single mostimportant parameter to determine (as the other two are generally more straightforward todetermine – see Section 2 and Kontar et al. 2011). For example, with δ = 4 (typical duringthe peak time of strong flares), a factor of 2 error inEc yields a factor of 4 error inPnth. Forlargerδ (as found in small flares, or rise/decay phases of large flares), such an error quicklyleads to an order of magnitude (or even greater) difference in the injected powerPnth and inthe total energy in the non-thermal electrons accelerated during the flare!

3.2 Why is the low-energy cutoff difficult to determine?

Fig. 3.1 Typical full-Sun flare spectrum.Dashed:Nonthermal thick-target spectrum from an acceleratedelectron distribution withδ=4, and a low-energy cutoff of 20 keV.Dotted:Thermal spectrum, from a plasmawith temperatureT = 20 MK and emission measureEM = 1049 cm−3. Solid: Total radiated spectrum. Themultiple peaks in the thermal spectrum are from spectral lines, as observed by an instrument with∼ 1 keVspectral resolution.

The essence of the problem in many flare spectra is summarizedin Figure 3.1: the non-thermal power-law is well-observed above∼20 keV, but any revealing features that it mightpossess at lower energies, such as a low-energy cutoff, are masked by the thermal emission.

Even if a spectrum does show a flattening at low energies that could be the result ofa low-energy cutoff, other mechanisms that could produce the flattening must be ruled out(see Section 3.4). The low-energy cutoff has the characteristic feature, determined by thephoton energy dependence of the bremsstrahlung cross-section (see Equation 2.9), that theX-ray spectrum eventually approaches a spectral index ofγ ≈ 1 at low energies (cf. Holman2003). It is currently impossible, however, to observe a flare spectrum to low enough photon

12 Holman et al.

energies to see that it does indeed become this flat. Generally we can only hope to rule outthe other mechanisms based on additional data and detailed spectral fits.

3.3 What is the shape of the low-energy cutoff, and how does itimpact the photonspectrum andPnth?

Bremsstrahlung photon spectra are obtained from convolution integrals over the electronflux distribution (equations 2.1 and 2.8). Hence, features in an electron distribution aresmoothed out in the resulting photon spectrum (see, e.g., Brown et al. 2006).

Fig. 3.2 Different shapes of low-energy cutoff in the injected electron distribution(left) lead to slightlydifferent photon spectra(right). The cutoff/turnover electron energy isEc=20 keV. The thin curve in the rightpanel demonstrates how the cutoff can be masked by emission from thermal plasma. See also Holman (2003)for a thorough discussion of bremsstrahlung spectra generated from electron power-laws with cutoff.

As can be seen in Figure 3.2, both a sharp cutoff atEc and a “turnover” (defined here tobe a constantF0(E) belowEc, a “plateau”) in the injected electron distribution lead tosimilarthick-target photon spectra. This subtle difference is difficult to discriminate observationally,and the problem is compounded by the dominance of the thermalcomponent at low energies.

A sharp cutoff would lead to plasma instabilities that should theoretically flatten thedistribution around and below the cutoff within microseconds (see Section 6). On the otherhand, the electron flux distribution below the cutoff must beflatter thanE−1, as demon-strated by Equation 3.1, or the total electron number flux would be infinite. Having a constantvalue for the distribution belowEc (turnover case) seems like a reasonable middle groundand approximates a quasilinearly relaxed electron distribution (Section 6; Krall & Trivelpiece1973, Chapter 10). Coulomb collisional losses, on the otherhand, yield an electron distri-bution that increases linearly at low energies (see Figure 3.3), leading to a photon spectrumbetween the sharp cutoff case and the turnover case.


Notice that the photon spectra actually flatten gradually tothe spectral index of 1 at lowenergies from the spectral index ofγ = δ +1 atEc and higher energies. BelowEc, it is nota power-law. Fitting a double power-law model photon spectrum, and using the break (i.e.,kink) energy as the low-energy cutoff typically leads to a large error inEc (e.g., Gan et al.2001; Saint-Hilaire & Benz 2005), and hence to an even largererror inPnth.

In terms of the energetics, Saint-Hilaire & Benz (2005) haveshown that the choice ofan exact shape for the low-energy cutoff as a model is not dramatically important. For afixed cutoff energy, from Equation 3.2 it can be shown that theratio of the power in theturnover model to the power in the sharp cutoff model withoutthe flat component belowthe cutoff energy isδ/2. In obtaining spectral fits, however, the turnover model gives highercutoff energies than the sharp cutoff model. Using simulations, Saint-Hilaire & Benz (2005)found that assuming either a sharp cutoff model or a turnovermodel led to differences inPnth typically less than∼20%. Hence, the sharp cutoff, being the simplest, is the model ofchoice for computing flare energetics. Nevertheless, knowing the shape of the low-energycutoff would not only yield more accurate non-thermal energy estimates, but would be asource of information on the acceleration mechanism and/orpropagation effects.

Fig. 3.3 The four plots show the Coulomb-collisional evolution withcolumn density of an injected electrondistribution (thick, solid line). For the simple power-law case (upper left), the low-energy end of the distri-bution becomes linear, and the peak of the distribution is found atEpeak= E∗/

√δ , whereδ is the injected

distribution power-law spectral index (δ=4 in the plots), andE∗ =√

2K ·N∗ is the initial energy that electronsmust possess in order not to be fully stopped by a column density N∗ (Equation 2.4). When a low-energycutoff is present, the peak of the distribution is seen to first decrease in energy untilE∗ exceeds the cutoffenergy (from Saint-Hilaire 2005).

Spectral inversion methods have recently been developed for deducing themean electronflux distribution(Equation 2.5) from X-ray spectra (Johns & Lin 1992; Brown etal. 2003,2006). A spectral “dip” has been found just above the presumed thermal component in somededuced mean electron flux distributions that may be associated with a low-energy cutoff

14 Holman et al.

(e.g., Piana et al. 2003). In the collisional thick-target model, the slope of the high-energy“wall” of this dip should be linear or flatter, with a linear slope indicating the absence ofemitting electrons in the injected electron distribution at the energies displaying this slope.Kontar & Brown (2006a) have found evidence for slopes that are steeper than linear, buttheir spectra were not corrected for photospheric albedo (see Section 3.5). Finding andunderstanding these dips is a crucial element for gaining anunderstanding of the low-energyproperties of flare electron distributions (see Kontar et al. 2011).

Emslie (2003) has pointed out that the non-thermal electrondistribution could seam-lessly merge into the thermal distribution, removing the need for a low-energy cutoff. Aswas shown by Holman et al. (2003) for SOL2002-07-23T00:35 (X4.8), however, merger ofthe electron distribution into the typically derived∼10–30 MK thermal flare plasma gener-ally implies an exceptionally high energy in non-thermal electrons. Thus, for a more likelyenergy content, a higher low-energy cutoff or a hotter plasma would need to be present inthe target region. Any emission from this additional “hot core,” because of its much loweremission measure, is likely to be masked by the usual∼10–30 MK thermal emission. Thismerger of the non-thermal electron distribution into the thermal tail in the target region doesnot remove the need for a low-energy cutoff in the electrons that escape the accelerationregion, however.

This section has dealt with the shape of the low-energy cutoff under the assumptions thatthe X-ray photon spectra are not altered by other mechanismsand that the bremsstrahlungemission is isotropic. The next section lists the importantcaveats to these assumptions, andtheir possible influence in the determination of the low-energy cutoff to the electron fluxdistribution.

3.4 Important caveats

As previously discussed, apparently minor features in the bremsstrahlung photon spectrumcan have substantial implications for the mean electron fluxand, consequently, the injectedelectron distribution. This means that unknown or poorly-understood processes that alterthe injected electron distribution (propagation effects,for example) or the photon spectrum(including instrumental effects) can lead to significant errors in the determination of thelow-energy cutoff. Known processes that affect the determination of the low-energy cutoffare enumerated below.

1. Detector pulse pileup effects (Smith et al. 2002), if not properly corrected for, can in-troduce a flattening of the spectrum toward lower energies that simulates the flatteningresulting from a low-energy cutoff.

2. The contribution of Compton back-scattered photons (photospheric albedo) to the mea-sured X-ray spectrum can simulate the spectral flattening produced by a low-energycutoff. Kasparova et al. (2005) have shown that the dip in aspectrum from SOL2002-08-20T08:25 (M3.4) becomes statistically insignificant when the spectrum is correctedfor photospheric albedo (also see Kontar et al. 2011). Kasparova et al. (2007) show thatspectra in the 15–20 keV energy band tend to be flatter near disk center when albedofrom isotropically emitted photons is not taken into account, further demonstrating theimportance of correcting for photospheric albedo.

3. The assumed differential cross-section and electron energy loss rate can influence theresults (for a discussion of this, see Saint-Hilaire & Benz 2005). In some circumstances,a contribution from recombination radiation may significantly change the results (Brownet al. 2010; also see Kontar et al., 2011).


4. Anisotropies in the electron beam directivity and the bremsstrahlung differential cross-section can significantly alter the X-ray spectrum (Massoneet al. 2004).

5. Non-uniform target ionization (the fact that the chromosphere’s ionization state varieswith depth, see Section 4) can introduce a spectral break that may be confused with thebreak associated with a low-energy cutoff.

6. Energy losses associated with a return current produce a low-energy flattening of theX-ray spectrum (Section 5). This is a low-energy “cutoff” inthe electron distributioninjected into the thick target, but it is produced between the acceleration region and theemitting source region.

7. A non-power-law distribution of injected electrons or significant evolution of the in-jected electron distribution during the observational integration time could affect thededuced value of the low-energy cutoff.

For all the above reasons, the value of the low-energy cutoffin the injected electron fluxdistribution has not been determined with any degree of certainty except perhaps in a fewspecial cases. Even less is known about the shape of the low-energy cutoff. The consensusin the solar physics community for now is to assume the simplest case, a sharp low-energycutoff. Existing studies, presented in the next section, tend to support the adequacy of thisassumption for the purposes of estimating the total power and energy in the acceleratedelectrons.

3.5 Determinations ofEc and electron energy content from flare data

Before RHESSI, instruments did not cover well (if at all) the∼10–40 keV photon ener-gies where the transition from thermal emission to non-thermal emission usually occurs.Researchers typically assumed an arbitrary low-energy cutoff at a value at or below the in-strument’s observing range (one would talk of the “injectedpower in electrons aboveEc

keV” instead of the total non-thermal powerPnth). An exception is Nitta et al. (1990). Theyargued that spectral flattening observed in two flares with the Solar Maximum MissionandHinotori indicated a cutoff energy of&50 keV. Also, Gan et al. (2001) interpreted spec-tral breaks at∼80 keV inCompton Gamma-Ray Observatory (CGRO)flare spectra as thelow-energy cutoff in estimating flare energetics, resulting in rather small values for the non-thermal energy in the analyzed flares. The relatively low resolution of the spectra from theseinstruments prevented the quantitative evaluation of any spectral flattening toward lowerenergies, however.

The only high-resolution flare spectral data before the launch of RHESSIwas the bal-loon data of Lin et al. (1981) for SOL1980-06-27T16:17 (M6.7) along with∼25 microflaresobserved during the same balloon flight. Benka & Holman (1994) applied a direct electricfield electron acceleration model to the SOL1980-06-27 flaredata. They derived, along withother model-related parameters, the time evolution of the critical energy above which run-away acceleration occurs – the model equivalent to the low-energy cutoff. They found thiscritical energy to range from∼20–40 keV.

It is now possible in most cases to obtain a meaningful upper limit on Ec, thanks toRHESSI’s high-spectral-resolution coverage of the 10–40 keV energy range and beyond.Holman et al. (2003), Emslie et al. (2004), and Saint-Hilaire & Benz (2005), in determiningthe low-energy cutoff, obtained the “highest value forEc that still fits the data.” In many solarflare spectra, because of the dominance of radiation from thermal plasma at low energies,a range of values forEc fit the data equally well, up to a certain critical energy, above

16 Holman et al.

which theχ2 goodness-of-fit parameter becomes unacceptably large. Thelow-energy cutoffis taken to be equal to this critical value. This upper limit on the cutoff energy results in alower limit for the non-thermal power and energy. The results obtained for the maximumvalue of Ec were typically in the 15–45 keV range, although late in the development ofSOL2002-07-23T00:35 (X4.8) some values as high as∼80 keV were obtained forEc. Theminimum non-thermal energies thus determined were comparable to or somewhat largerthan the calculated thermal energies.

Fig. 3.4 RHESSIspatially integrated spectra in four time intervals duringSOL2002-04-15T03:55 (M1.2).(a) Spectrum at 23:06:20–23:06:40 UT (early rise phase). (b) Spectrum at 23:09:00-23:09:20 UT (just beforeimpulsive phase). (c) Spectrum at 23:10:00–23:10:20 UT (soon after the impulsive rise). (d ) Spectrum at23:11:00–23:11:20 UT (at the hard X-ray peak). The plus signs with error bars represent the spectral data.The lines represent model spectral fits: the dashed lines arenon-thermal thick-target bremsstrahlung, thedotted lines are thermal bremsstrahlung, and the solid lines are the summation of the two (from Sui et al.2005a).

One of the best determinations of the low-energy cutoff so far was obtained by Sui et al.(2005a). They complemented the spatially-integrated spectral data for the SOL2002-04-15T03:55 (M1.2) limb flare with imaging and lightcurve information. Four spectra fromthis flare are shown in Figure 3.4. The earliest spectrum, before the impulsive rise of thehigher energy X-rays, was well fitted with an isothermal model. The last spectrum, fromthe time of the hard X-ray peak, clearly shows a thermal component below∼20 keV. Of


particular interest is the second spectrum, showing both thermal and non-thermal fit com-ponents. As a consequence of the flattening of the isothermalcomponent at low energies,the low-energy cutoff to the non-thermal component cannot extend to arbitrarily low ener-gies without exceeding the observed emission. This places atight constraint on the valueof the low-energy cutoff. The additional requirement that the time evolution of the derivedtemperature and emission measure of the thermal component be smooth and continuousthroughout the flare constrains the value at other times. Applying the collisional thick-targetmodel with a power-law distribution of injected electrons,they found the best cutoff value tobeEc = 24±2 keV (roughly constant throughout the flare). The energy associated with thesenon-thermal electrons was found to be comparable to the peakenergy in the X-ray-emittingthermal plasma, but an order of magnitude greater than the kinetic energy of the associ-ated coronal mass ejection (CME) (Sui et al. 2005b). This contrasts with results obtainedfor large flares, where the minimum energy in non-thermal electrons is typically found to beless than or on the order of the energy in the CME (e.g., Emslieet al. 2004).

The importance of correcting for the distortion of spectra by albedo was revealed by asearch for low-energy cutoffs in a sample of 177 flares with relatively flat spectra (γ ≤ 4)between 15 and 20 keV (Kontar et al. 2008a). Spectra can be significantly flattened by thepresence of albedo photons in this energy range. The X-ray spectra, integrated over theduration of the impulsive phase of the flares, were inverted to obtain the correspondingmean electron flux distributions. Eighteen of the flares showed significant dips in the meanelectron flux distribution in the 13-19 keV electron energy range that might be associatedwith a low-energy cutoff (see Section 3.3). However, when the X-ray spectra were correctedfor albedo from isotropically emitted X-rays, all of the dips disappeared. Therefore, theauthors concluded that none of these flare electron distributions had a low-energy cutoffabove 12 keV, the lowest electron energy in their analysis.

Low-energy cutoffs were identified in the spectra of a sampleof early impulsive flaresobserved byRHESSIin 2002 (Sui et al. 2007). Early impulsive flares are flares in whichthe>25 keV hard X-ray flux increase is delayed by less than 30 s after the flux increase atlower energies. The pre-impulsive-phase heating of plasmato X-ray-emitting temperaturesis minimal in these flares, allowing the nonthermal part of the spectrum to be observedto lower energies. In the sample of 33 flares, 9 showed spectral flattening at low energiesin spectra obtained throughout the duration of each flare with a 4 s integration time. Aftercorrecting for the albedo from isotropically emitted X-rays, the flattening in 3 of the 9 flares,all near Sun center, disappeared. The flattening that persisted in the remaining 6 flares wasconsistent with that produced by a low-energy cutoff. The values derived for the low-energycutoff ranged from 15 to 50 keV. The authors found the evolution of the spectral break andthe corresponding low-energy cutoff in these flares to be correlated with the non-thermalhard X-ray flux. Further studies are needed to assess the significance of this correlation.

Low-energy cutoffs with values exceeding 100 keV were identified in the spectra of thelarge flare SOL2005-01-19T08:22 (X1.3) (Warmuth et al. 2009). The hard X-ray light curveof this flare consisted of multiple peaks that have been interpreted as quasi-periodic oscilla-tions driven by either magnetoacoustic oscillations in a nearby loop (Nakariakov et al. 2006)or by super-Alfvenic beams in the vicinity of the reconnection region (Ofman & Sui 2006).The high low-energy cutoffs were found in the last major peakof the series of hard X-raypeaks. Unlike the earlier peaks, this peak was also unusual in that it was not accompaniedby the Neupert effect (see Section 8.3), consistent with thehigh values of the low-energycutoff, and it exhibited soft-hard-harder rather than soft-hard-soft spectral evolution (seeSection 9.1). A change in the character of the observed radioemission and movement of oneof the two hard X-ray footpoints into a region of stronger photospheric magnetic field were

18 Holman et al.

also observed at the time of this peak. These changes suggesta strong connection betweenlarge-scale flare evolution and electron acceleration.

4 Nonuniform ionization in the thick-target region

In the interpretation of hard X-ray (HXR) spectra in terms ofthe thick-target model, oneeffect which has been largely ignored until recently is thatof varying ionization along thepath of the thick-target beam. As first discussed by Brown (1973), the decrease of ioniza-tion with depth in the solar atmosphere reduces long-range collisional energy losses. Thisenhances the HXR bremsstrahlung efficiency there, elevating the high energy end of theHXR spectrum by a factor of up to 2.8 above that for a fully ionized target. The net re-sult is that a power-law electron spectrum of indexδ produces a photon spectrum of indexγ = δ −1 at low and high energies (see Equation 2.12), but withγ < δ −1 in between. Theupward break, where the spectrum begins to flatten toward higher energies, occurs at fairlylow energies, probably masked in measured spectra by the tail of the thermal component.The downward knee, where the spectrum steepens again toγ = δ − 1, occurs in the fewdeka-keV range, depending on the column depth of the transition zone. Thus, the measuredX-ray spectrum may show a flattening similar to that expectedfor a low-energy cutoff in theelectron distribution.

4.1 Electron energy losses and X-ray emission in a nonuniformly ionized plasma

The collisional energy-loss cross-sectionQc(E) is dependent on the ionization of the back-ground medium. Flare-accelerated electron beams can propagate in the fully ionized coronaas well as in the partially ionized transition region and chromosphere.. Following Hayakawa & Kitao(1956) and Brown (1973), the cross-sectionQc(E) can be written for a hydrogen plasmaionization fractionx

Qc(E) =2πe4

E2 (xΛee+(1−x)ΛeH) =2πe4

E2 Λ(x+λ ), (4.1)

wheree is the electronic charge,Λee the electron-electron logarithm for fully ionized mediaandΛeH is an effective Coulomb logarithm for electron-hydrogen atom collisions. Numeri-cally Λee≃ 20 andΛeH ≃ 7.1, soΛ = Λee−ΛeH ≃ 12.9 andλ = ΛeH/Λ ≃ 0.55.

Then, in a hydrogen target of ionization levelx(N) at column densityN(z), the energyloss equation for electron energyE is (cf. Equation 2.2)

dEdN

=−2πe4ΛE

(λ +x(N)) =−K′

E(λ +x(N)), (4.2)

whereK′ = 2πe4Λ = (Λ/Λee)K ≃ 0.65K.The energy loss of a given particle with initial energyE0 depends on the column density

N(z) =∫ z

0 n(z′)dz′, so the electron energy at a given distancez from the injection site can bewrittenE2 = E2

0 −2K′M(N(z)) (cf. Equation 2.4), where

M(N(z)) =

N(z)∫

0

(λ +x(N′))dN′ (4.3)


Fig. 4.1 Photon spectrum residuals, normalized by the statistical error for the spectral fit, for the time interval00:30:00 – 00:30:20 UT, 2002-July-23, for (upper panel) an isothermal Maxwellian plus a power-law and(lower panel) an isothermal Maxwellian plus the nonuniform ionization spectrum withδ = 4.24 andE∗ =53 keV (from Kontar et al. 2003).

is the “effective” ionization-weighted collisional column density.The fractional atmospheric ionizationx as a function of column densityN (cm−2) changes

from 1 to near 0 over a small spatial range in the solar atmosphere. Therefore, to lowestorder,x(N) can be approximated by a step functionx(N) = 1 for N < N∗, andx(N) = 0for N ≥ N∗. This givesM(N) = (λ +1)N for N < N∗ andM(N) = N∗+λN for N ≥ N∗.Electrons injected into the target with energies less thanE∗ =

√

2K′(λ +1)N∗ =√

2KN∗experience energy losses and emit X-rays in the fully ionized plasma withx = 1, as in thestandard thick-target model. Electrons injected with energies higher thanE∗ lose part oftheir energy and partially emit X-rays in the un-ionized (x= 0), or, more generally, partiallyionized plasma.

We can deduce the properties of the X-ray spectrum by substituting Equation 4.2 intoEquation 2.7 (withdN= nvdt) and comparingIthick(ε) from Equation 2.6 withIthick(ε) fromEquation 2.8. We see that for the nonuniformly ionized case the denominator in the innerintegral now containsλ + x(N) andK is replaced withK′. In the step-function model forx(N), photon energies greater than or equal toε∗ = E∗ are emitted by electrons in the un-ionized plasma withE≥E∗. Sinceλ +x(N) has the constant valueλ , the thick-target power-law spectrum is obtained (for injected power-law spectrum), but the numerical coefficientcontainsK′λ = 2πe4ΛeH instead ofK. At photon energies far enough belowε∗ that thecontribution from electrons withE ≥ E∗ is negligible,λ +x(N) = λ +1 and the numericalcoefficient contains(λ + 1)K′ = K. The usual thick-target spectral shape and numericalcoefficient are recovered. The ratio of the amplitude of the high-energy power-law spectrumto the low-energy power-law spectrum is(λ +1)/λ ≃ 2.8. The photon energyε∗(keV) ≃2.3×10−9

√

N∗(cm−2), between where the photon spectrum flattens below the high-energypower law and above the low-energy power law, determines thevalue of the column densitywhere the plasma ionization fraction drops from 1 to 0.

4.2 Application to flare X-ray spectra

The step-function nonuniform ionization model was used by Kontar et al. (2002, 2003) tofit photon spectra from five flares. They assume a single power-law distribution of injected

20 Holman et al.

electrons with power-law indexδ and approximate the bremsstrahlung cross-section withthe Kramers cross-section. First, they fit the spectra to thesum of a thermal Maxwellianat a single temperatureT plus a single power law of indexγ . For SOL2002-07-23T00:35(X4.8) (Kontar et al. 2003) they limit themselves to deviations from a power law in the non-thermal component of the spectrum above∼40 keV. The top panel of Figure 4.1 shows anexample of such deviations, which represent significant deviations from the power-law fit.These deviations are much reduced by replacing the power lawwith the spectrum from thenonuniform ionization model, with the minimum rms residuals obtained for values ofδ =4.24 andE∗ = 53 keV (Figure 4.1, bottom panel). The corresponding minimum (reduced)χ2 value obtained for the best fit to the X-ray spectrum (10–130 keV) dropped from 1.4 forthe power-law fit to 0.8 for the nonuniform ionization fit. There are still significant residualspresent in the range from 10 to 30 keV; these might be due to photospheric albedo or theassumption of a single-temperature thermal component.

By assuming that the main spectral feature observed in a hardX-ray spectrum is dueto the increased bremsstrahlung efficiency of the un-ionized chromosphere, allowance fornonuniform target ionization offers an elegant direct explanation for the shape of the ob-served hard X-ray spectrum and provides a measure of the location of the transition re-gion. Table 4.1 shows the best fit parameters derived for the four flare spectra analyzed byKontar et al. (2002). The last column shows the ratio of the minimum χ2 value obtainedfrom the nonuniform ionization fit to the minimumχ2 value obtained from a uniform ion-ization (single power-law) fit to the non-isothermal part ofthe spectrum. The nonuniformionization model fits clearly provide substantially betterfits than single power-law fits.

Table 4.1 Best fit nonuniformly ionized target model parameters for a single power-lawF0(E0), the equiv-alentN∗ (energy range 20-100 keV), and the ratio ofχ2

nonuni/χ2uni (from Kontar et al. 2002)

.

Date Time, UT kT(keV) δ E∗ (keV) N∗ (cm2) χ2nonuni/χ2

uni20 Feb 2002 11:06 1.47 5.29 37.4 2.7×1020 0.03217 Mar 2002 19:27 1.27 4.99 24.4 1.1×1020 0.04731 May 2002 00:06 2.02 4.15 56.2 6.1×1020 0.041

1 Jun 2002 03:53 1.45 4.46 21.0 8.4×1019 0.055

Values of the fit parameterskT (keV), δ andE∗ as a function of time for SOL2002-07-23T00:35 (X4.8) , together with the corresponding value ofN∗(cm−2) ≃ 1.9×1017E∗(keV)2

were obtained by Kontar et al. (2003). The results (Figure 4.2) demonstrate that the thermalplasma temperature rises quickly to a value≃ 3 keV and decreases fairly slowly thereafter.The injected electron flux spectral indexδ follows a general “soft-hard-soft” trend and qual-itatively agrees with the time history of the simple best-fitpower-law indexγ (Holman et al.2003).E∗ rises quickly during the first minute or so from∼40 keV to∼70 keV near theflare peak and thereafter declines rather slowly. The corresponding values ofN∗ are∼2-5×1020 cm−2.

The essential results of these studies are that (1) for a single power-law electron injectionspectrum, the expression for bremsstrahlung emission froma nonuniformly-ionized targetprovides a significantly better fit to observed spectra than the expression for a uniform target;and (2) the value ofE∗ (and correspondinglyN∗) varies with time.

An upper limit on the degree of spectral flattening∆γ that can result from nonuniformionization was derived by Su et al. (2009). They applied thisupper limit to spectra from asample of 20 flares observed byRHESSIin the period 2002 through 2004. They found that


Fig. 4.2 Variation of kT, δ , E∗, and N∗ throughout SOL2002-07-23T00:35 (X4.8) (Kontar et al. 2003).The variation of other parameters, such as emission measure, can be found in Holman et al. (2003) andCaspi & Lin (2010).

15 of the 20 flare spectra required a downward spectral break at low energies and for each ofthese 15 spectra derived the difference∆γ of the best-fit power-law spectral indices aboveand below the break. A Monte Carlo method was used to determine the 95% confidenceinterval for each of the derived values of∆γ . Taking the value of∆γ to be incompatible withnonuniform ionization if the 95% confidence interval fell above the derived upper limit,Su et al. (2009) found that six of the flare spectra could not beexplained by nonuniformionization alone. Thus, for these six flares some other causesuch as a low-energy cutoff orreturn-current-associated energy losses (Section 5) mustbe at least partially responsible forthe spectral flattening.

5 Return current losses

The thick-target model assumes that a beam of electrons is injected at the top of a loop and“precipitates” downwards in the solar atmosphere. Unless accompanied by an equal flux ofpositively charged particles, these electrons constitutea current and must create a significant

22 Holman et al.

self-induced electric field that in turn drives a co-spatialreturn current for compensation(Hoyng et al. 1976; Knight & Sturrock 1977; Emslie 1980; D’Iakonov & Somov 1988). Thereturn current consists of ambient electrons, plus any primary electrons that have scatteredback into the upward direction. By this means we have a full electric circuit of precipitatingand returning electrons that keeps the whole system neutraland the electron beam stableagainst being pinched off by the self-generated magnetic field required by Ampere’s law foran unneutralized beam current. However, the self-induced electric field results in a potentialdrop along the path of the electron beam that decelerates and, therefore, removes energyfrom the beam electrons.

5.1 The return current electric field

The initial formation of the beam/return-current system has been studied by van den Oord(1990) and references therein. We assume here that the system has time to reach a quasi-steady state. Van den Oord finds this time scale to be on the order of the thermal electron-ioncollision time. This time scale is typically less than or much less than one second, depend-ing on the temperature and density of the ambient plasma. In numerical simulations bySiversky & Zharkova (2009), times to reach a steady state after injection ranged from 0.07 sto 0.2 s, depending on the initial beam parameters.

The self-induced electric field strength at a given locationzalong the beam and the flareloop, E (z), is determined by the current density associated with the electron beam,j(z),and the local conductivity of the loop plasma,σ (z), through Ohm’s law:E (z) = j(z)/σ (z).Relating the current density to the density distribution function of the precipitating electrons,f (z,E,θ), whereE is the electron energy andθ is the electron pitch angle, gives

E (z) =2√

2πσ (z)

e√me

1∫

0

∞∫

0

f (z,E,θ)√

EµdEdµ . (5.1)

Here µ is the cosine of the pitch angle ande and me are the electron charge and mass,respectively. The self-induced electric field strengthE (z) depends on the local distributionof the beam electrons, which in turn depends on the electric field already experienced bythe beam as well as any Coulomb energy losses and pitch-anglescattering that may havesignificantly altered the beam. It also depends on the local plasma temperature (and, to alesser extent, density) throughσ (z), which can, in turn, be altered by the interaction of thebeam with the loop plasma (i.e., local heating and “chromospheric evaporation”). Therefore,determination of the self-induced electric field and its impact on the precipitating electronsgenerally requires self-consistent modeling of the coupled beam/plasma system.

Such models have been computed by Zharkova & Gordovskyy (2005, 2006). They nu-merically integrate the time-dependent Fokker-Planck equation to obtain the self-inducedelectric field strength and electron distribution functionalong a model flare loop. The in-jected electron beam was assumed to have a single power-law energy distribution in theenergy range fromElow = 8 keV toEupp = 384 keV and a normal (Gaussian) distribution inpitch-angle cosineµ with half-width dispersion∆ µ = 0.2 aboutµ = 1.

The model computations show that the strength of the self-induced electric field is nearlyconstant at upper coronal levels and rapidly decreases withdepth (column density) in thelower corona and transition region. The rapidity of the decrease depends on the beam fluxspectral index. It is steeper for softer beams (δ=5-7) than for harder ones (δ=3). The strengthof the electric field is higher for a higher injected beam energy flux density (erg cm−2 s−1),

Electron Acceleration and Propagation 23P

ho

ton

Flu

xP

ho

ton

Flu

x

Photon Energy (keV)

Photon Energy (keV)

δ

γ

log (F0 / 1 erg / sq cm / s)

γhig

h - γl

ow

(a)

(b)

(c)

(d)

Fig. 5.1 (a) Photon spectra computed from full kinetic solutions including return current losses and colli-sional losses and scattering. The top spectrum is for an injected single-power-law electron flux distributionbetween 8 keV and 384 keV with an index ofδ = 3, and the bottom spectrum is forδ = 7. The injectedelectron energy flux density is 108 erg cm−2 s−1. (b) Same as (a), but for an injected energy flux density of1012 erg cm−2 s−1. The tangent lines at 20 and 100 keV demonstrate the determination of the low-energy andhigh-energy power-law spectral indicesγlow andγhigh. (c) The photon spectral indicesγlow (dashed lines) andγhigh (solid lines) vs.δ for an injected energy flux density of 108 (squares), 1010 (circles), and 1012 erg cm−2

s−1 (crosses). (d)γhigh− γlow vs. the log of the injected electron energy flux density forδ equal to 3 (bottomcurve, squares), 5 (middle curve, circles), and 7 (top curve, triangles) (from Zharkova & Gordovskyy 2006).

and the distance from the injection point over which the electric field strength is highest (andnearly constant) decreases with increasing beam flux density.

5.2 Impact on hard X-ray spectra

Deceleration of the precipitating beam by the electric fieldmost significantly affects thelower energy electrons (<100 keV), since the fraction of the original particle energylostto the electric field is greater for lower energy electrons. This leads to flattening of theelectron distribution function towards the lower energiesand, therefore, flattening of thephoton spectrum.

Photon spectra computed from kinetic solutions that include return current energy lossesand collisional energy losses and scattering are shown in Figure 5.1 (a) and (b). Low- andhigh-energy spectral indices and their dependence on the power-law index of the injectedelectron distribution and on the injected beam energy flux density are shown in Figure 5.1(c) and (d). The difference between the high-energy and low-energy spectral indices is seen

24 Holman et al.

to increase with both the beam energy flux density and the injected electron power-law indexδ . The low-energy index is found to be less than 2 forδ as high as 5 when the energy fluxdensity is as high as 1012 erg cm−2 s−1.

5.3 Observational evidence for the presence of the return current

We have seen that return current energy losses can introducecurvature into a spectrum,possibly explaining the “break” often seen in observed flareX-ray spectra. A difficulty indirectly testing this explanation is that the thick-targetmodel provides the power (energyflux) in the electron beam (erg s−1), but not the energy flux density (erg cm−2 s−1). X-rayimages provide information about the area of the target, butthis is typically an upper limiton the area. Even if the source area does appear to be well determined, the electron beamcan be filamented so that it does not fill the entire area (the filling factor is less than 1).Also, if only an upper limit on the low-energy cutoff to the electron distribution is known, asdescribed in Section 3.5, the energy flux density may be higher. Therefore, the observationstypically only give a lower limit on the beam energy flux density.

The non-thermal hard X-ray flux is proportional to the electron beam flux density, butthe return-current energy losses are also proportional to the beam flux density. As a conse-quence, Emslie (1980) concluded that the flux density of the non-thermal X-ray emissionfrom a flare cannot exceed on the order of 10−15 cm−2 s−1 above 20 keV. Alexander & Daou(2007) have deduced the photon flux density from non-thermalelectrons in a sample of 10flares ranging fromGOESclass M1.8 to X17. They find that the non-thermal photon fluxdensity does not monotonically increase with the thermal energy flux, but levels off (satu-rates) as the thermal energy flux becomes high. They argue that this saturation most likelyresults from the growing importance of return current energy losses as the electron beam fluxincreases to high values in the larger flares. They find that the highest non-thermal photonflux densities agree with an upper limit computed by Emslie.

A correlation between the X-ray flux and spectral break energy was found by Sui et al.(2007) in their study of X-ray spectra in early impulsive flares (see Section 3.5). They pointout that the increasing impact of return current energy losses on higher energy electrons asthe electron beam energy flux density increases could be an explanation for this correlation.

Battaglia & Benz (2008) studied two flares with non-thermal coronal hard X-ray sourcesfor which the difference between the measured photon spectral index at the footpoints andthe spectral index of the coronal source was greater than two, the value expected for coronalthin-target emission and footpoint thick-target emissionfrom a single power-law electrondistribution (see Section 10.2). They argue that return-current losses between the coronaland footpoint source regions are most likely responsible for the large difference between thespectral indices.

The return current can also affect the spectral line emission from flares. Evidence forthe presence of the return current at the chromospheric level from observations of the linearpolarization of the hydrogen Hα and Hβ lines has been presented by Henoux & Karlicky(2003). Dzifcakova & Karlicky (2008) have shown that the presence of a return current inthe corona may have a distinguishable impact on the relativeintensities of spectral linesemitted from the corona.


6 Beam-plasma and current instabilities

Interaction of the accelerated electrons with plasma turbulence as they stream toward thethick-target emission region can modify the electron distribution. In this section we brieflydiscuss a likely source of such turbulence: that generated by the electron beam itself. If thebeam is or becomes unstable to driving the growth of plasma waves, these waves can interactwith the beam and modify its energy and/or pitch angle distribution until the instabilityis removed or the wave growth is stabilized. The return current associated with the beam(Section 5) can also become unstable, resulting in greater energy loss from the beam. Beam-plasma and return-current instabilities in solar flares have been reviewed by Melrose (1990)and Benz (2002).

A sharp, low-energy cutoff to an electron beam or, more generally, a positive slope inthe beam electron energy distribution is well-known to generate the growth of electrostaticplasma waves (Langmuir waves). The characteristic time scale for the growth of these wavesis on the order of[(Nb/n)ωpe]

−1, whereωpe is the electron plasma frequency andNb/n theratio of the density of unstable electrons in the beam to the plasma density. This is on theorder of microseconds for a typical coronal loop plasma density and Nb/n ≈ 10−3. Thisplasma instability is often referred to as the bump-on-tailinstability. The result is that ona somewhat longer but comparable time scale electrons from the unstable region of theelectron distribution lose energy to the waves until the sharp cutoff is flattened so that thedistribution no longer drives the rapid growth of the waves.Therefore, the electron energydistribution below the low-energy cutoff is likely to rapidly become flat or nearly flat (suffi-ciently flat to stabilize the instability) after the electrons escape the acceleration region (seeChapters 9 & 10 of Krall & Trivelpiece 1973).

A recent simulation of the bump-on-tail instability for solar flare conditions, includingCoulomb collisions and wave damping as the electrons propagate into an increasingly denseplasma, has been carried out by Hannah et al. (2009). The authors compute the mean elec-tron flux distribution in their model flare atmosphere after injecting a power-law distributionwith a sharp low-energy cutoff. They find a mean electron flux distribution with no dip (seeSection 3.3) and a slightly negative slope below the original cutoff energy with a spectralindexδ between 0 and 1.

A beam for which the mean electron velocity parallel to the magnetic field substantiallyexceeds the mean perpendicular velocity can drive the growth of waves that resonantly in-teract with the beam. When the electron gyrofrequency exceeds the plasma frequency, thesewaves are electrostatic and primarily scatter the electrons in pitch angle. Generally knownas the anomalous Doppler resonance instability, this instability tends to isotropize the beamelectrons. Holman et al. (1982) showed that under solar flareconditions this instability cangrow and rapidly isotropize the beam electrons in less than amillisecond. They found thatelectrons at both the low- and high-energy ends of the distribution may remain unscattered,however, because of wave damping. This could result in up to two breaks in the emitted X-ray spectrum. On the other hand, Vlahos & Rowland (1984) haveargued that non-thermaltails will form in the ambient plasma and stabilize the anomalous Doppler resonance insta-bility by suppressing the growth of the plasma waves.

Electrons streaming into a converging magnetic field can develop a loss-cone distribu-tion, with a deficit of electrons at small pitch angles. Both classical Coulomb collisionsand loss-cone instabilities can relax this distribution byscattering electrons into the losscone or extracting energy from the component of the electronvelocities perpendicular to themagnetic field (e.g., Aschwanden 1990). One loss-cone instability, the electron-cyclotron

26 Holman et al.

or gyrosynchrotron maser, produces coherent radiation observable at radio frequencies(Holman et al. 1980; Melrose & Dulk 1982).

The return current associated with the streaming electronsbecomes unstable to the ion-acoustic instability when its drift speed exceeds a value onthe order of the ion sound speed.The excited ion sound waves enhance the plasma resistivity,increasing the electric fieldstrength associated with the return current, the heating ofthe plasma by the current, and theenergy loss from the electron beam.

It has been argued that rapid plasma heating and particle acceleration in the coronashould result in the expansion of hot plasma down the legs of flare loops at the ion soundspeed, confined behind a collisionless ion-acoustic conduction front (Brown et al. 1979).Electrons with speeds greater than about three times the electron thermal speed would beable to stream ahead of the conduction front. This scenario has not been observationallyverified, but the observational signature may be confused bythe chromospheric evaporationproduced by the high-energy particles streaming ahead of the conduction front.

Rowland & Vlahos (1985) argued that if the electron beam is unstable to beam plasmainteractions, the return current will be carried by high-velocity electrons. This reduces theimpact of collisions on the beam/return-current system andhelps stabilize the system. In arecent simulation, Karlicky et al. (2008) have found that for current drift velocities exceed-ing the electron thermal speed, the return current is carried by both the primary (driftingthermal) current and an extended tail of high-velocity electrons.

The evolution of the electron-beam/return-current systemwhen the return-current driftspeed exceeds the electron thermal speed has also been simulated by Lee et al. (2008). Theyfind that double layers form in the return current, regions ofenhanced electric field thatfurther increase the energy losses of the electron beam. This, in turn, increases the high-est electron energy to which these losses significantly flatten the electron distribution andcorresponding hard X-ray spectrum.

The beam/return-current system has been simulated by Karlicky (2009), with a focus onthe role of the Weibel instability. The Weibel instability tends to isotropize the electron dis-tribution. Karlicky & Kasparova (2009) have computed the thin-target X-ray emission fromthe evolved electron distributions for a model with a weak magnetic field and another modelwith a strong magnetic field (ratio of the electron gyrofrequency to the plasma frequency∼0 and∼ 1, respectively). They demonstrate that in both cases the electron distribution ismore isotropic and the directivity of the X-ray emission is lower than when the instability ofthe system is not taken into account, with the greatest isotropization occurring in the weakfield limit.

Although we expect plasma instabilities to affect the evolution of the electron beam,observationally identifying them is difficult. The bump-on-tail instability and return currentlosses both lead to a flat low-energy cutoff. So far we have notestablished the ability toobservationally distinguish a flat low-energy cutoff from asharp low-energy cutoff. Thebump-on-tail instability may be distinguishable from return-current losses by its short timescale and, therefore, the short distance from the acceleration region over which it effectivelyremoves the unstable positive slope from the electron energy distribution. The instabili-ties that isotropize the electron pitch-angle distribution may be responsible for evidencefrom albedo measurements that flare electron distributionsare isotropic or nearly isotropic(Kontar & Brown 2006b).


7 Height dependence and size of X-ray sources with energy andtime

7.1 Footpoint Sources

Hard X-ray footpoint sources result from collisional bremsstrahlung radiated by precipitat-ing electrons, which produce most of the emission in the chromosphere according to thecollisional thick-target model. Depending on the density structure in the legs of the coronalmagnetic loop, mildly energetic electrons lose their energy in the lower corona or transi-tion region, while the more energetic electrons penetrate deeper into the chromosphere (seeEquation 2.2).

The altitude of these hard X-ray footpoint sources could never be measured accuratelybeforeRHESSI, because of a lack of spatial and spectral resolution. WithRHESSI, we canmeasure the centroid of the footpoint location with an accuracy of order an arcsecond forevery photon energy in steps as small as 1 keV. For a flare near the limb (Figure 7.1), thecentroid location translates directly into an altitude.

Aschwanden et al. (2002) studied such a flare, SOL2002-02-20T11:07 (C7.5). The heightsof the footpoint sources were fitted with a power-law function of the photon energy. Thisyielded altitudesh ≈ 1000−5000 km in the energy rangeε = 10−60 keV, progressivelylower with higher energy, as expected from the thick-targetmodel (Figure 7.1, right frame).

Since the stopping depth of the precipitating electrons is afunction of column density,the integrated density along their path in the chromosphere(equation 2.4), the measuredheight dependence of the hard X-ray centroids can be inverted to yield a density modelof the chromosphere (Brown et al. 2002). Assuming the decrease in density with heighthad a power-law dependence and the plasma is fully ionized, the inversion of theRHESSIdata in the example shown in Figure 7.1 yielded a chromospheric density model that hasa significantly higher electron density in theh= 2000−5000 km range than the standardchromospheric models based on UV spectroscopy and hydrostatic equilibrium (VAL andFAL models). TheRHESSI-based chromospheric density model was therefore found tobe more consistent with the “spicular extended chromosphere,” similar to the results fromsub-mm radio observations during solar eclipses carried out at Caltech (Ewell et al. 1993).

Forward fittingRHESSIX-ray visibilities to an assumed circular Gaussian source shape,Kontar et al. (2008b) found for a limb flare the full width at half maximum (FWHM) sizeand centroid positions of hard X-ray sources as a function ofphoton energy with a claimedresolution of∼0′′.2. They show that the height variation of the chromosphericdensity andof the magnetic flux density can be found with a vertical resolution of∼150 km by mappingthe 18−250 keV X-ray emission of energetic electrons propagating in the loop at chromo-spheric heights of 400−1500 km. Assuming collisional losses in neutral hydrogen with anexponential decrease in density with height, their observations of SOL2004-01-06T06:29(M5.8) suggest that the density of the neutral gas is in good agreement with hydrostaticmodels with a scale height of around 140±30 km. FWHM sizes of the X-ray sources de-crease with energy, suggesting the expansion (fanning out)of magnetic flux tubes in thechromosphere with height. The magnetic scale heightB(z)(dB/dz)−1 is found to be on theorder of 300 km and a strong horizontal magnetic field is associated with noticeable fluxtube expansion at a height of∼900 km. A subsequent analysis with an assumed ellipticalGaussian source shape (Kontar et al. 2010) confirms these results and shows that the verti-cal extent of the X-ray source decrease with increasing X-ray energy. The authors find thevertical source sizes to be larger than expected from the thick-target model and suggest thata multi-threaded density structure in the chromosphere is required. The thick-target model

28 Holman et al.

890 900 910 920

250

260

270

280

10 20 30 40 50 60Energy ε[keV]

-1

0

1

2

3

4

5

Alti

tude

h3(

ε) [M

m]

10 20 30 40 50 60Energy ε[keV]

-1

0

1

2

3

4

5

Alti

tude

h3(

ε) [M

m]

h(ε)=r(ε)-r0=h0*(ε/20 keV)a

r0=674.227 Mmh0= 2.273 Mma =-1.32χ2= 0.50

Fig. 7.1 The centroids of footpoint hard X-ray emission are marked for different photon energies between 10keV and 60 keV for SOL2002-02-20T11:07 (C7.5), which occurred near the solar west limb and was imagedwith RHESSI(left panel). The altitudeh(ε) as a function of energyε shows a systematic height decrease withincreasing energy (right panel) (from Aschwanden et al. 2002).

to which the results were compared, however, did not accountfor partial occultation of theX-ray sources by the solar limb.

The flare SOL2002-02-20T11:07 (C7.5) has been reanalyzed byPrato et al. (2009) us-ing both photon maps over a range of photon energies and mean electron flux maps deducedfrom RHESSIvisibilities over a range of electron energies. Using source centroids computedfrom the maps and assuming an exponential decrease in density with height, they found thedensity scale height to be an order of magnitude larger than the expected chromosphericscale height on the quiet Sun, but consistent with the scale height in a non-static, flaringatmosphere. This is also consistent with the enhanced plasma densities found at∼1000-5000 km altitudes by Aschwanden et al. (2002).

If the results for the 400-1500 km height range (Kontar et al.2008b) and for the∼1000-5000 km height range (Aschwanden et al. 2002; Prato et al. 2009) are typical of flare loops,they imply that the upper chromosphere and transition region respond with a non-hydrostatic,expanded atmosphere while the low chromosphere does not respond to the flare energy re-lease. These results could, of course, depend on the magnitude of the flare. More studies ofthis kind are clearly desirable, especially in coordination with observations of spectral linesfrom the chromosphere and transition region.

7.2 Loop Sources and their Evolution

As discussed above, footpoint sources are produced by bremsstrahlung emission in the thick-target chromosphere. The compactness of such sources results from the rapid increase of thedensity from the tenuous corona to the much denser chromosphere. This also gives riseto the compact height distribution of emission centroids atdifferent energies as shown inFigure 7.1. However, if the density distribution has a somewhat gradual variation, one wouldexpect a more diffuse height distribution. Specifically, atsome intermediate energies, weexpect that HXR emission would appear in the legs of the loop,rather than the commonlyobserved looptop sources at low energies and footpoint sources at high energies. This has


Fig. 7.2 CLEAN images at 04:58:22-04:58:26 UT during the impulsive phase of SOL2003-11-13T05:01(M1.6). The background shows the image at 9-12 keV. The contour levels are 75% and 90% of the peak fluxat 9-12 keV (looptop), 70% and 90% at 12–18 keV (legs), and 50%, 60%, & 80% at 28–43 keV (footpoints)(from Liu et al. 2006).

been observed byRHESSIin SOL2003-11-13T05:01 (M1.6) (Liu et al. 2006) (Figure 7.2)and in SOL2002-11-28T04:37 (C1.0) (Sui et al. 2006b).

To reveal more details of the energy-dependent structure ofSOL2003-11-13T05:01(M1.6), Figures 7.3a-c show the X-ray emission profile along the flare loop at differentenergies for three time intervals in sequence. The high energy emission is dominated by thefootpoints, but there is a decrease of the separation of the footpoints with decreasing energyand with time. At later times the profile becomes a single source, peaking at the looptop.The general trend suggests an increase of the plasma densityin the loop with time (Liu et al.2006), which can be produced by chromospheric evaporation and can give rise to progres-sively shorter stopping distances for electrons at a given energy. Such a density increase alsosmooths out to some extent the sharp density jump at the transition region. This results inthe non-thermal bremsstrahlung HXRs at intermediate energies appearing in the legs of theloop, at higher altitudes than the footpoints, as shown in Figure 7.2.

From the emission profiles in the non-thermal regimes of the photon spectra, Liu et al.(2006) derived the density distribution along the loop, using the empirical formula for non-thermal bremsstrahlung emission profiles given by Leach & Petrosian (1983, Equation 11).Leach and Petrosian found that this formula closely approximates their numerical resultsfor a steady-state, power-law injected electron distribution with a uniform pitch-angle dis-tribution, no return-current losses, and a loop with no magnetic field convergence. Since thisformula is a function of the column density, one does not needto assume any model form ofthe density distribution (cf. Aschwanden et al. 2002). Figure 7.4 shows the density profilesderived from the emission profiles in the three time intervals shown in Figure 7.3. Between

30 Holman et al.

Fig. 7.3 (a) Brightness profiles in different energy bands measured along a semi-circular path fit to the flaringloop for the time interval 04:58:00–04:58:24 UT of SOL2003-11-13T05:01. The vertical axis indicates theaverage photon energy (logarithmic scale) of the energy band for the profile. Representative energy bands(in units of keV) are labeled above the corresponding profiles. The filled circles mark the local maxima, andthe vertical dotted lines are the average positions of the centroids of the looptop and footpoint sources. (b,c) Same as (a), but for 04:58:24–04:58:48 and 04:58:48–04:59:12 UT, respectively. The error bars show theuncertainty of the corresponding profile (from Liu et al. 2006).

Fig. 7.4 Averaged density profiles along one leg of the loop inferred from the HXR brightness profilesduring the three time intervals in Figure 7.3. The distance is measured along the leg extending from thecentroid of the thermal looptop source at about 15 arcsec in Figure 7.3 to the end of the fitted semi-circle atabout 37 arcsec (from Liu et al. 2006).

the first and second intervals, the density increases dramatically in the lower part of the loop,while the density near the looptop remains essentially unchanged. The density enhancementthen shifts to the looptop from the second to the third interval. This indicates a mass flowfrom the chromosphere to the looptop, most likely caused by chromospheric evaporation.For papers studying chromospheric evaporation using coordinatedRHESSIHXR and EUVDoppler-shift observations, see Milligan et al. (2006a,b)and Brosius & Holman (2007).


Fig. 7.5 RHESSI(solid lines) andGOES1-8 A (dotted line) light curves are shown in thetop panel. TheRHESSIenergy bands (from top to bottom) are 3–6, 6–12, 12–25, and 50–100 keV, with scaling factors of 5,1, 4, 3, and 0.5, respectively. TheRHESSIandGOESintegration times are 4 and 3 s, respectively. Thebottompanelshows the distance between the 3-6 keV moving source centroids and their corresponding footpointcentroids located in the 25-50 keV image of the flare at the time of peak emission. The distances are plane-of-sky values with no correction for motions away from or toward the observer (from Sui et al. 2006b).

The flare SOL2002-11-28T04:37 (C1.0) was an early impulsiveflare, meaning that therewas minimal pre-heating of plasma to X-ray-emitting temperatures prior to the appear-ance of impulsive hard X-ray emission (see Section 3.5).RHESSIobservations of this flareshowed coronal X-ray sources that first moved downward and then upward along the legsof the flare loop (Sui et al. 2006b). The bottom panel of Figure7.5 shows the motion of thesources observed in the 3-6 keV band.RHESSIandGOESlight curves are shown in the toppanel. The sources originated at the top of the flare loop and then moved downward alongboth legs of the loop until the time of peak emission at energies above 12 keV. Afterward thesource in the northern leg of the loop was no longer observable, but the source in the south-ern leg moved back to the top of the loop. Its centroid location at the looptop was slightlybut significantly lower than the centroid position at the beginning of the flare. Higher-energysources showed a similar evolution, but they had lower centroid positions than their lowerenergy counterparts, again in agreement with the predictions of the thick-target model.

The early downward source motion along the legs of the loop isa previously unobservedphenomenon. At this time we do not know if the occurrence is rare, or if it is simply rarelyobserved because of masking by the radiation from the thermal plasma. Sui et al. (2006b)argue that the motion results from the hardening of the X-rayspectrum, and possibly an in-crease in the low-energy cutoff, as the flare hard X-ray emission rises to its peak intensity. Aflatter spectrum results in a higher mean energy of the electrons contributing to the radiationat a given X-ray energy. In a loop with a plasma density that increases significantly from thetop to the footpoints, these higher energy electrons will propagate to a lower altitude in the

32 Holman et al.

loop as the spectrum hardens. The softening of the spectrum after peak emission would alsocontribute to the upward motion of the source after the peak.However, at that time chromo-spheric evaporation would likely be increasing the densityin the loop, as discussed abovefor SOL2003-11-13T05:01 (M1.6) , and thermal emission would be more important. All ofthese can contribute to an increase in the height of the centroid of the X-ray source. Thedownward motion may only occur in initially cool flare loops,i.e., early impulsive flares,because these loops are most likely to contain the density gradients that are required.

In an attempt to differentiate between thermal and non-thermal X-ray emission, Xu et al.(2008) modeled the size dependence with photon energy of coronal X-ray sources observedby RHESSIin ten M-class limb flares. They determined the one-sigma Gaussian width of thesources along the length of the flare loops by obtaining forward fits to the source visibilities.The integration times ranged from one to ten minutes and the source sizes were determinedin up to eight energy bins ranging in energy from as low as 7 keVto as high as 30 keV. Theyfound the source sizes to increase slowly with photon energy, on average asε1/2. The resultswere compared with several models for the variation of the source size with energy. Thesource size was expected to vary asε−1/2 for a thermal model with a constant loop densityand a temperature that decreased with a Gaussian profile along the legs of the loop froma maximum temperature at the top of the loop. For the injection of a power-law electronflux distribution into a high-density loop so that the loop isa collisional thick target, thesource size was expected to increase asε2. Neither of these models are consistent with theobservedε1/2 dependence. A hybrid thermal/non-thermal model and a non-thermal modelwith an extended acceleration region at the top of the loop were found to be consistent withthe deduced scaling, however. The extended acceleration region was deduced to have a half-length in the range 10′′ – 18′′ and density in the range(1−5)×1011 cm−3. We note that theextended acceleration region model implies a column density in the range 0.73−6.5×1020

cm−2 along the half length and, from Equation 2.4, all electrons with energies less thansomewhere in the range of 23 keV – 68 keV that traverse this half length will lose all of theirenergy to collisions. The acceleration process would therefore need to be efficient enough toovercome these losses. On the other hand, the 7 – 30 keV energyrange is the range in whichfits to spatially integrated X-ray spectra typically show a combination of both thermal andnon-thermal bremsstrahlung emission.

Studies of flare hard X-ray source positions and sizes as a function of photon energy andtime hold great promise for determining the height structure of flare plasma and its evolution,as well as providing information about the magnetic structure of the flare loop. Such studiesare currently in their early stages, in that they usually assume an over simplified power-lawor exponential height distribution for the plasma and do nottake into account the variationof the plasma ionization state with height. They also assumethe simple, one-dimensionalcollisional thick-target model, without consideration ofthe pitch-angle distribution of thebeam electrons or the possibility of additional energy losses to the beam (such as return-current losses). Given the potential for obtaining a betterunderstanding of flare evolution,we look forward to the application of more sophisticated models to the flare hard X-ray data.

8 Hard X-ray timing

The analysis of energy-dependent time delays allows us to test theoretical models of physi-cal time scales and their scaling laws with energy. In the wavelength domain of hard X-rays,there are at least three physical processes known in the observation of solar flares that leadto measurable time delays as a function of energy (for a review, see Aschwanden 2004): (1)


time-of-flight dispersion of free-streaming electrons, (2) magnetic trapping with the colli-sional precipitation of electrons, and (3) cooling of the thermal plasma.

8.1 Time-of-Flight Delays

The first type, thetime-of-flight (TOF)delay, has a scaling of∆ t(ε) ∝ ε−1/2 and is causedby velocity differences of electrons that propagate from the coronal acceleration site to thechromospheric energy-loss region. The time differences are of order∆ t ≈ 10−100 ms fornon-thermal electrons at energiesE ≈ 20−100 keV (e.g., Aschwanden et al. 1995, 1996).The measurement of such tiny time delays requires high photon statistics and high timeresolution. Such data were provided byCGRO/BATSE, which had 8 detectors, each withan effective collecting area of∼2000 cm2 and oriented at different angles to the Sun so thatdetector saturation at high count rates was not a problem. (For comparison, the total effectivecollecting area ofRHESSI’s detectors is less than 100 cm2.)

These studies of TOF delays have provided important evidence that electrons are ac-celerated in the corona, above the top of the hot flare loops observed in soft X-rays. Thefine structure in the light curves of most, but not all, of the studied flare bursts showedenergy-dependent time delays consistent with the free streaming of electrons to the foot-points of the flare loops from an origin somewhat more distantthan the half-length of theloops (Aschwanden et al. 1995; Aschwanden & Schwartz 1995; Aschwanden et al. 1996).

8.2 Trapping Delays

The second type, thetrapping delay, is caused by magnetic mirroring of coronal electronswhich precipitate toward the chromosphere after a collisional time scale∆ t(ε) ∝ ε3/2. Thisis observable for time differences of∆ t ≈ 1−10 s for non-thermal electrons atE ≈ 20−100 keV (e.g., Vilmer et al. 1982; Aschwanden et al. 1997). For trapping delays the higherenergy X-rays lag the lower energy X-rays, as opposed to time-of-flight delays where thehigher energy X-rays precede the lower energy X-rays.

Aschwanden et al. (1997) filtered variations on time scales∼1 s or less out ofCGROBATSE flare HXR light curves. They found time delays in the remaining gradually vary-ing component to be consistent with magnetic trapping and collisional precipitation of theparticles. Trap plasma densities∼ 1011 cm−3 were deduced. No evidence was found for adiscontinuity in the delay time as a function of energy and, therefore, for second-step (two-stage) acceleration of electrons at energies≤ 200 keV.

8.3 Thermal Delays

The third type, thethermal delay, can be caused by the temperature dependence of cool-ing processes, such as by thermal conduction,τc(T) ∝ T−5/2 (e.g., Antiochos & Sturrock1978; Culhane et al. 1994), or by radiative cooling,τr(T) ∝ T5/3 (e.g., Fisher & Hawley1990; Cargill et al. 1995). The observed physical parameters suggest that thermal conduc-tion dominates in flare loops at high temperatures as observed in soft X-ray wavelengths,while radiative cooling dominates in the later phase in postflare loops as observed in EUVwavelengths (Antiochos & Sturrock 1978; Culhane et al. 1994; Aschwanden & Alexander

34 Holman et al.

2001). When the temperature drops in the decay phase of flares, the heating rate can jus-tifiably be neglected and the conductive or radiative cooling rate dominate the temperatureevolution. BeforeRHESSI, the cooling curveT(t) in flare plasmas had been studied in onlya few flares (e.g., McTiernan et al. 1993; Culhane et al. 1994;Aschwanden & Alexander2001).

The high spectral resolution ofRHESSIdata is particularly suitable for any type of ther-mal modeling, because we can probe the thermal plasma from∼3 keV up to∼30 keVwith a FWHM resolution of∼1 keV thanks to the cooled germanium detectors (Lin et al.2002; Smith et al. 2002). This allows us to measure flare temperatures with more confidence.A statistical study of flare temperatures measured in the range of T ≈ 7−20 MK indeeddemonstrates some agreement between the values obtained from spectral fitting ofRHESSIdata with those obtained fromGOESflux ratios (Battaglia et al. 2005), althoughRHESSIhas a bias for the high-temperature tail of the differentialemission measure (DEM) distri-bution (Aschwanden et al. 2008; Vaananen & Pohjolainen 2007). Of course, we expect anagreement between the deduced emission-measure-weightedtemperatures only when bothinstruments are sensitive to a temperature range that covers the flare DEM peak.

A close relationship between the non-thermal and thermal time profiles was found earlyon, in the sense that the thermal emission often closely resembles the integral of the non-thermal emission, a relationship that is now known as theNeupert effect(Neupert 1968;Hudson 1991; Dennis & Zarro 1993). This relationship is, however, strictly only expectedfor the asymptotic limit of very long cooling times, while a physically more accurate modelwould quantify this effect by a convolution of the non-thermal heating with a finite coolingtime. The deconvolution of the e-folding cooling time in such a model has never been at-tempted statistically and as a function of energy or temperature. Theoretical discussions ofthe Neupert effect, including multiple energy release events, chromospheric evaporation, andcooling, can be found in Warren & Antiochos (2004), Liu et al.(2010), and Reeves & Moats(2010).

The cooling time at a given energy can be estimated from the decay time of a flaretime profile. For instance, the decay times measured withGOESin soft X-rays were foundto have a median ofτdecay≈ 6 min (Veronig et al. 2002a,b). The observed cooling timeshave typically been found to be much longer than predicted from classical conduction,but shorter than the radiative cooling time (e.g., McTiernan et al. 1993; Jiang et al. 2006;Raymond et al. 2007). This discrepancy could result from either continuous heating or sup-pression of conduction during the decay phase, or a combination of both.

The Neupert effect was tested by correlating the soft X-ray peak flux with the (time-integrated) hard X-ray flux. A high correlation and time coincidence between the soft X-raypeak and hard X-ray end time was generally found, but a significant fraction of events alsohad a different timing (Veronig et al. 2002c). A delay of 12 s was found in the soft X-rayflux time derivative with respect to the hard X-ray flux in SOL2003-11-13T05:01 (M1.6)(Liu et al. 2006, also see Section 7.2). Time delays such as this could be related to the hy-drodynamic flow time during chromospheric evaporation. Tests of the “theoretical Neuperteffect,” i.e., comparisons of the beam power supply of hard X-ray-emitting electrons and thethermal energy of evaporated plasma observed in soft X-rays, found it to strongly dependon the low-energy cutoff to the non-thermal electron distribution (Veronig et al. 2005). Thisprovides another approach to deducing the energy at which the low-energy cutoff in theelectron distribution occurs in individual flares. The Neupert effect has also been studied inseveral flares by Ning (2008, 2009), who finds a high correlation between the hard X-rayflux and the time derivative of the thermal energy deduced from X-ray spectral fits (Ning


1 10 100Energy ε[keV]

10-14

10-12

10-10

10-8

10-6

Mu

lti-

the

rma

l sp

ect

rum

I(ε

) [

ph

oto

ns

s-1 c

m-2

ke

V-1

]

1.3

15

5.8 keV

εx [keV] =

1.7

20

7.8 keV

2.2

25

9.7 keV

2.6

30

11.6 keV

3.0

35

13.6 keV

3.5

40

15.5 keV

3.9

45

17.5 keV

4.3

50

19.4 keV

Eth [keV]

T [MK]

dEM(T)/dT~T - 4.0

I( ε)~ε- 3.5

Fig. 8.1 Example of a multi-thermal spectrum with contributions from plasmas with temperatures ofT =15,20, ...,50 MK and a DEM distribution ofdEM(T)/dT ∝ T−4. The individual thermal spectra and theirsum are shown with thin linestyle, where the sum represents the observed spectrum. Note that the photonsin the energy rangeε = 5.8− 19.4 keV are dominated by temperatures of T=15–50 MK, which haveacorresponding thermal energy that is about a factor of(4+1/2) = 4.5 lower than the corresponding photonenergy (εth = 1.3−4.3 keV). The summed photon spectrum without the high-temperature cutoff approachesthe power-law functionI(ε) ∝ ε−3.5 (dotted line) (from Aschwanden 2007).

2008) and an anti-correlation between the hard X-ray spectral index and the time rate ofchange of the UV flare area observed byTRACE(Ning 2009).

8.4 Multi-Thermal Delay Modeling withRHESSI

Since major solar flares generally produce a large number of individual postflare loops, giv-ing the familiar appearance of loop arcades lined up along the flare ribbons, it is unavoidablethat each loop is heated up and cools down at different times,so that a spatially integratedspectrum always contains a multi-thermal differential emission measure distribution (cf.Warren 2006). The resulting multi-thermal bremsstrahlungspectrum (for photon energies

36 Holman et al.

26-Feb-2002 10:25:52-26-Feb-2002 10:31:36

0 50 100 150 200 250 300Time after start t-t0[s]

0

500

1000

1500

2000

2500

3000C

ount

s pe

r s

file=demod_05.savdur_flare= 344 st1,t2= 20, 90 sdt= 0.5 stsmo= 10.0 sch= 1-19E =11-50 keVx,y= 920,-232

05

10 20 30 40 50 100Photon energy ε [keV]

1

10

100

Pho

tons

/s c

m2 k

eV

γnth= 2.0γth = 6.3Enth=30-50 keVEth=16.6 keVδEM = 6.6qe= 6.8

Eth Enth Emax

20 30 40 50 60 70 80 90Time after start t-t0[s]

0.0

0.2

0.4

0.6

0.8

1.0

1.2

Cou

nts

(nor

mal

ized

)

FWHM

FWHM= 27.5 stGauss= 11.7 sEnth=30 keV

-5 0 5 10 15 20Time delay ∆t-<∆tnth> [s]

10

20

30

40

50

100

HX

R e

nerg

y [k

eV]

Eth

Enth

Emax

tc0= 301 sdtnth= 0.04 sβ= 2.8α= 6.1σt= 0.3 sχ= 1.1

Fig. 8.2 X-ray light curves are shown for SOL2002-02-26T10:27 (C9.6), for energies of 10 keV to 30 keVin intervals of 1 keV, observed withRHESSI(left panels). The spectrum is decomposed into thermal and non-thermal components (top right panel) and the delay of the peaks at different energies is fitted with a thermalconduction cooling time model that has a scaling ofτcond(T)≈ T−β (right bottom panel). The best fit showsa power index ofβ = 2.8, which is close to the theoretically expected value ofβ = 5/2 (Equation 8.2). Thefull delay of the thermal component is indicated with a thin curve (bottom left panel), while the weighted(thermal+non-thermal) fit is indicated with a thick curve (from Aschwanden 2007).

ε) observed in soft X-rays (neglecting the Gaunt factor of order unity),

I(ε) = I0

∫

exp(−ε/kBT)

T1/2

dEM(T)dT

dT , (8.1)

is then a function of a multi-thermaldifferential emission measuredistributiondEM(T) =n2(T)dV. An example of a multi-thermal spectrum from a differentialemission measureproportional toT−4 up to a maximum temperature of 50 MK is shown in Figure 8.1.

As discussed above, the initial cooling of the hot flare plasma (say atT∼>10 MK) isgenerally dominated by conductive cooling (rather than by radiative cooling, which candominate later after the plasma cools to EUV-emitting temperatures ofT∼<2 MK). The


thermal conduction time has the following temperature dependence:

τcond(T) =εth

dE/dtcond=

3nekBTddsκT5/2 dT

ds

≈ 212

neL2kB

κT−5/2 = τc0

(

TT0

)−5/2

; (8.2)

see Aschwanden (2007) for parameter definitions. Since the thermal bremsstrahlung at de-creasing photon energies is dominated by radiation from lower temperature flare plasma, theconductive cooling time is expected to become longer at lower temperatures (τcond∝ T−5/2).Thus, the soft X-ray peak is always delayed with respect to the harder X-ray peaks, reflectingthe conductive cooling of the flare loops.

Aschwanden (2007) has measured and modeled this conductivecooling delayτcond(ε)for a comprehensive set of short-duration (≤ 10 min) flares observed byRHESSI. One ex-ample is shown in Figure 8.2. He finds that the cooling delay∆ t expressed as a function ofthe photon energyε and photon spectral indexγ can be approximated by

∆ t(ε ,γ)≈ τg74

[

log

(

1+τc0

τg

(

ε(γ −1)ε0

)−β)]3/4

, (8.3)

(whereτg is the Gaussian width of the time profile peak) and yields a newdiagnostic ofthe process of conductive cooling in multi-thermal flare plasmas. In a statistical study of 65flares (Aschwanden 2007), 44 (68%) were well fit by the multi-thermal model with a best fitvalue for the exponent ofβ = 2.7±1.2, which is consistent with the theoretically expectedvalue ofβ = 2.5 according to Equation 8.2. The conductive cooling time atT0 = 11.6 MK(ε0 = 1 keV) was found to range from 2 to 750 s, with a mean value ofτc0 = 40 s.

We note that these timing data, as well as thick-target fits tothe non-thermal part of spec-tra that reveal the evolution of the energy content in accelerated electrons, provide additionalconstraints on models such as the multithread flare model of Warren (2006).

9 Hard X-ray spectral evolution in flares

9.1 Observations of spectral evolution

The non-thermal hard X-ray emission from solar flares, best observed in the 20 to 100 keVrange, is highly variable. Often several emission spikes with durations ranging from secondsto minutes are observed. In larger events, sometimes a more slowly variable, long durationemission can be observed in the later phase of the flare. Hence, most flares start out with animpulsivephase, while some events, mostly large ones, show the presence of a lategradualphase in the hard X-ray time profile.

While these two different behaviors can already be spotted by looking at lightcurves,they also are distinct in their spectral evolution. The impulsive spikes tend to be harder atthe peak time, and softer both in the rise and decay phase. Thespectrum starts soft, getsharder as the flux rises and softens again after the maximum ofthe emission. This pattern ofthe spectral evolution is thus calledsoft-hard-soft(SHS). On the other hand, in the gradualphase, the flux often slowly decreases, while the spectrum stays hard or gets even harder.This different kind of spectral evolution is calledsoft-hard-harder(SHH).

Historically, both the SHS (Parks & Winckler 1969; Kane & Anderson 1970) and theSHH behavior (Frost & Dennis 1971) were observed in the earlyera of hard X-ray observa-tions of the Sun. Subsequent investigation confirmed both the SHS (Benz 1977; Brown & Loran

38 Holman et al.

Fig. 9.1 Time evolution of the spectral indexγ (upper curve, linear scale on right) and the flux normalizationI35 (lower curve, logarithmic scale on left) of the non-thermal component in SOL2002-11-09T13:23 (M4.9).Different emission spikes are shown in different colors (after Grigis & Benz 2004).

1985; Lin & Schwartz 1987; Gan 1998; Fletcher & Hudson 2002; Hudson & Farnık 2002)and the SHH (Cliver et al. 1986; Kiplinger 1995; Saldanha et al. 2008; Grigis & Benz 2008)behavior. The SHH behavior has been found to be correlated with proton events in interplan-etary space (Kiplinger 1995; Saldanha et al. 2008; Grayson et al. 2009).

Evidence for hard-soft-hard (HSH) spectral evolution at energies above∼50 keV hasbeen reported for multiple spikes in SOL2004-11-03T03:35 (M1.6) (Shao & Huang 2009b).SHS behavior was observed at lower energies. This HSH behavior might be explained byalbedo, which typically peaks around 30–40 keV (see Kontar et al. 2011), but the authorscorrected for albedo from isotropically emitted photons. Alikely explanation is that thespikes overlie a harder, gradually varying component, possibly emission from trapped elec-trons (Section 8.2).

While all these observations established the qualitative properties of the spectral evo-lution, a statistical analysis of the quantitative relation between the flux and spectral indexhad not been performed in the pre-RHESSIera. Here, we summarizeRHESSIresults inves-tigating quantitatively the spectral evolution of the nonthermal component of the hard X-rayemission, as well as the theoretical implications. More details can be found in Grigis & Benz(2004, 2005, 2006).

To quantify the spectral evolution, a simple parameterization for the shape of the non-thermal spectrum is needed. Luckily, in solar flares the spectrum is well described by apower law in energy, which often steepens above 50 keV. Such asoftening of the spectrumcan be modeled by a broken power-law model. However, it is difficult to observe such a


downward bending at times of weak flux, because the high-energy region of the spectrumis lost in the background. As a compromise, Grigis & Benz (2004) fitted the data to a singlepower-law function at all times. Although the single power law does not always providea good fit to the spectra, it provides a characteristic spectral slope and ensures a uniformtreatment of the spectra at different times.

The two free parameters of the power-law model are the spectral indexγ and the power-law normalizationIε0 at the reference energyε0. The reference energyε0 is arbitrary, butfixed, usually near the logarithmic mean of the covered energy range. In theRHESSIspectralanalysis software, OSPEX,ε0 = 50 keV by default. The time dependent spectrum is givenby

I(ε , t) = Iε0(t)

(

εε0

)−γ(t). (9.1)

A representative sample of 24 solar flares ofGOESmagnitudes between M1 and X1 wasselected by Grigis & Benz (2004). The spectral model (Equation 9.1), with the addition ofan isothermal emission component at low energies, was fittedwith a cadence of oneRHESSIspin period (about 4 s). This delivered a sequence of measurements of the quantitiesIε0(t)andγ(t) for each of the 24 events, covering a total time of about 62 minutes of non-thermalhard X-ray emission. For these events,ε0 = 35 keV was chosen, a meaningful energy whichlies about in the middle of the range where the non-thermal emission is best observed inthese M-class flares.

An example of the measured time evolution of the spectral index γ and the flux nor-malizationI35 for the longer-lasting event of the set is shown in Figure 9.1. A correlationin time between the two curves can be readily seen. Single emission spikes are plotted indifferent colors, so that the soft-hard-soft evolution canbe observed during each spike (withthe exception of the late, more gradual phase, where the emission stays hard as the fluxdecays).

As there is an anti-correlation in time between logI35(t) andγ(t), a plot of one parameteras a function of the other, eliminating the time dependence,shows the relationship betweenthem. Figure 9.2 shows plots ofγ vs. I35 for 3 events where there are only one or twoemission peaks. The points in the longer uninterrupted riseor decay phase during each eventare marked by plus symbols. A linear relationship between logI35 andγ can be seen duringeach phase, although it can be different during rise and decay.

On the other hand, a plot of all the 911 fitted model parametersfor all the events show alarge scatter, as shown in Figure 9.3. The large scatter can be understood as originating fromthe superposition of data from a large numbers of different emission spikes, each featuringlinear trends with different parameters. This plot does demonstrate, however, the tendencyfor flatter spectra to be associated with more intense flares.

RHESSIobservations of the gradual phase of large solar flares (Grigis & Benz 2008) andits relation with proton events (Saldanha et al. 2008; Grayson et al. 2009) have shown thatthe hardening behavior is complex and cannot be characterized by a continuously increas-ing hardness during the event. Therefore the soft-hard-harder (SHH) denomination does notaccurately reflect the observed spectral evolution. Rather, phases of hardening (or even ap-proximatively constant hardness) are often seen in larger events as the flux decays (Kiplinger1995). The start of the hardening phase can happen near the main peak of the flare, or later.The end of hardening can even be followed by new impulsive SHSpeaks. The most re-cent statistical study of the correlation of SHH behavior with proton events (Grayson et al.2009) found that in a sample of 37 flares that were magnetically well-connected to Earth, 18

40 Holman et al.

Fig. 9.2 Spectral indexγ vs. flux normalizationI35 for three events, showing the linear dependence of singlerise and decay phases of emission spikes on a log-linear scale. Dots mark results from individual spikes, whilepluses mark the longer rise or decay phase (from Grigis & Benz2004).

showed SHH behavior and 12 of these produced solar energeticparticle (SEP) events. Noneof the remaining 19 flares that did not show SHH behavior produced SEP events.

9.2 Interpretation of spectral evolution

Can we explain the soft-hard-soft spectral behavior theoretically? The problem here is thatmany effects contribute to the properties of the high-energy electron distribution whosebremsstrahlung hard X-rays are observed byRHESSIand similar instruments. We can iden-tify three main, closely related classes of physical processes that affect the distribution ofthe electrons and the spectrum of the X-ray photons they generate: (1) theaccelerationofpart of the thermal ambient plasma, (2) theescapefrom the acceleration region, and (3) thetransportto the emitting region. The photon spectrum also depends on the properties of thebremsstrahlung emission mechanism.

Miller et al. (1996) proposed a stochastic acceleration mechanism where electrons areenergized by small-amplitude turbulent fast-mode waves, called the transit-time dampingmodel. They showed that their model could successfully account for the observed numberand energy of electrons accelerated above 20 keV in subsecond spikes or energy releasefragments in impulsive solar flares. However, they made no attempt to explain the observedhard X-ray spectra (which are softer than predicted by the transit-time damping model) anddid not consider spectral evolution. Furthermore, this approach does not account for particleescape. Grigis & Benz (2006) extended the model with the addition of a term describing theescape of the particles from the acceleration region, as in the model of Petrosian & Donaghy(1999). To ensure conservation of particles, they also add asource term of cold particlescoming into the accelerator (such as can be provided by a return current).


Fig. 9.3 Plot of the spectral indexγ versus the fitted non-thermal flux at 35 keV (given in photons s−1 cm−2

keV−1). All 911 data points from the 24 events are shown (from Grigis & Benz 2004).

The stochastic nature of this acceleration model implies that the electrons undergo adiffusion process in energy space. Mathematically, the acceleration is described by the fol-lowing convective-diffusive equation:

∂ f∂ t

=12

∂ 2

∂E2

[

(DCOLL +DT) f]

− ∂∂E

[

(ACOLL +AT) f]

−S(E) · f +Q(E) , (9.2)

where f (E) is the electron density distribution function,DT andAT are, respectively, thediffusion and convection coefficients due to the interactions of the electrons with the accel-erating turbulent waves,DCOLL andACOLL are, respectively, the diffusion and convection co-efficients due to collisions with the ambient plasma,S(E) is the sink (escape) term, andQ(E)is the source (return current) term. The escape term is proportional tov(E)/τ , wherev(E) isthe electron speed, andτ is the escape time. The escape time can be energy-dependent,butfor simplicity it is initially kept constant. The longer theescape time, the better the particlesare trapped in the accelerator. The source term is in the formof a Maxwellian distributionof electrons with the same temperature as the ambient plasma.

The coefficientsDT andAT are proportional to the dimensionless acceleration parameter

IACC =UT

UB· c〈k〉

ΩH, (9.3)

whereUT andUB are, respectively, the energy densities of the turbulent waves and of theambient magnetic field,〈k〉 is the average wave vector, andΩH is the proton gyrofrequency.

42 Holman et al.

1 10 100Energy (keV)

100

102

104

106

108

1010

Ele

ctro

n di

strib

utio

n (c

m-3ke

V-1)

δ=2

δ=3.5

δ=5

δ=7

Fig. 9.4 Accelerated electron density distributions with different values of the power-law index resultingfrom changes inIτ = IACC · τ . The dashed curve represents the ambient Maxwellian distribution. The twodotted lines indicate the energy range used for the computation of the power-law indexδ shown above eachspectrum. Harder spectra have a largerIτ value (from Grigis & Benz 2006).

Equation 9.2 can be solved numerically until an equilibriumstate (∂ f /∂ t =0) is reached.The equilibrium electron spectra from the model are controlled by two parameters: the ac-celeration parameterIACC described above and the escape timeτ . Above 10-20 keV, thecollision and source terms in Equation (9.2) can be neglected, since they apply to the am-bient Maxwellian, and thus the equilibrium spectra depend to a first approximation only onthe productIτ = IACC · τ .

Figure 9.4 shows the equilibrium electron spectra for different values ofIτ = IACC · τ .As Iτ increases, the spectrum gets harder and harder. To explain the soft-hard-soft effect,either the acceleration or the trapping efficiency (or both)must increase until the peak time,and then decrease again. We note that this model does not include magnetic trapping (otherthan in the magnetic turbulence itself), which can alter thecomputed electron spectra andtheir time evolution (e.g., Metcalf & Alexander 1999).

To see whether this produces the linear relation between thespectral index and the logof the flux normalization, Grigis & Benz (2006) computed the hard X-ray emission fromthese model electron spectra. Since these are equilibrium spectra, thin-target emission wascomputed. They then plotted the spectral index vs. the flux normalization of the resultingphoton spectra. Since the spectra are not power-law, but bend down, they fit a power-lawmodel to the model photon spectrum in a similar range as the one used for the observations.

Figure 9.5 shows the computed values for the spectral indices and flux normalizationsfor both the electron and the photon spectrum from the model.The results show that thereis indeed a linear relation between the spectral index and the log of the flux normalization.

An alternative mechanism that could be responsible for soft-hard-soft spectral evolutionis return current losses as the electrons propagate to and within the thick-target footpointsof the flare loop (Zharkova & Gordovskyy 2006). The highest electron energy to which re-


Fig. 9.5 Model results for the spectral index and flux normalization for electrons and photons. The dashedline is the best straight-line fit to the model results (in therange of spectral indices from 2 to 8 for the electrons,and 3 to 9 for the photons), corresponding to a pivot-point behavior (from Grigis & Benz 2006).

turn current losses are significant is proportional to the return current electric field strength,which is in turn proportional to the electron beam flux density (see Section 5). Therefore, asthe electron flux density increases and then decreases, the low-energy part of the X-ray spec-trum flattens to higher and then lower energies as the return current electric field strengthincreases and then decreases. The net effect is SHS spectralevolution below the maximumenergy for which return current losses are significant during the flare. The observation ofSHS behavior in coronal X-ray sources, however, indicates that this spectral evolution is aproperty of the acceleration mechanism rather than a consequence of energy losses duringelectron propagation (Battaglia & Benz 2006, see Section 10.3).

Are there two stages of electron acceleration, one responsible for the impulsive phaseand one for the gradual phase?RHESSIspectroscopy and imaging of a set of 5 flares withhardening phases (Grigis & Benz 2008) showed that there is nodiscontinuity in the motionof footpoints at the onset of hardening and no clear separation between the impulsive andthe gradual phase: the former seems to smoothly merge into the latter. This supports theview that the same acceleration mechanism changes gradually in the later phase of the flare,rather than a two stage acceleration theory. The hardening phase may in fact be caused byan increase in the efficiency of trapping of the electrons above 100 keV.

The underlying cause of the SHS spectral evolution has been addressed in terms of thestochastic acceleration model by Bykov & Fleishman (2009) and Liu & Fletcher (2009).Bykov and Fleishman consider acceleration in strong, long-wavelength MHD turbulence,taking into account the effect of the accelerated particleson the turbulence. They arguethat the electron spectrum flattens during the linear acceleration phase, while the spectrumsteepens during the nonlinear phase when damping of the turbulence because of the parti-cle acceleration is important, giving SHS spectral evolution. They argue that SHH evolu-

44 Holman et al.

tion will be observed when the injection of particles into the acceleration region is strong.Liu & Fletcher also argue that the SHS evolution results fromdependence of the electrondistribution power-law index on the level of turbulence as it increases and subsequentlydecreases. They attribute changes in the SHS correlation during a flare to changes in thebackground plasma, likely due to chromospheric evaporation.

We note that simple direct-current (DC) electric field acceleration of electrons out of thethermal plasma can produce the SHS spectral evolution. The flux of accelerated electronsand the maximum energy to which electrons are accelerated and, therefore, the high-energycutoff to the electron distribution, increase and decreasetogether as the electric field strengthincreases and decreases (Holman 1985). The X-ray spectrum is steeper at energies withinone to two orders of magnitude below the high-energy cutoff (Holman 2003). In large flares,however, where the X-ray spectrum is observed to continue toMeV energies or higher, thereis no evidence for a high-energy cutoff in the appropriate energy range. Therefore, at least forlarge flares with spectra extending to high energies, a simple DC electric field accelerationmodel does not appear to be appropriate.

10 The connection between footpoint and coronal hard X-ray sources

Hard X-ray (HXR) sources at both footpoints of a coronal loopstructure have been observedsince Hoyng et al. (1981). As reviewed in Sections 1 & 2, they are understood to be thick-target bremsstrahlung emission produced by precipitatingelectrons, accelerated somewherein or above the loop. A third HXR source situated above the looptop (see Krucker et al.2008a, for a review) was first noted by Masuda et al. (1994) inYohkohobservations. Thenature of this coronal HXR source has remained uncertain, but in simple solar flare modelswith reconnection and particle acceleration in the corona,we expect some relation betweencoronal HXR sources and footpoints.RHESSIhas enabled us to study events featuring coro-nal HXR sources and footpoints simultaneously. By studyingthe behavior of the sources intime and the relations between them, we can address questions such as: Are both coronaland footpoint emissions caused by the same electron population? How is such an electronbeam modified in the loop (collisions, return currents, trapping, etc.)? Is SHS behavior (Sec-tion 9.1) a transport effect produced by collisions or return currents, or is it a feature imposedby the acceleration mechanism?

10.1 RHESSIimaging spectroscopy

RHESSIhas provided the possibility of obtaining simultaneous, high-resolution imagedspectra at different locations on the Sun. One can thereforestudy each source separatelyin events with several contemporaneous HXR sources. The high spectral resolution has al-lowed a reliable differentiation between thermal and non-thermal emission to be made inmany flares. Furthermore,RHESSI’s imaging spectroscopy has allowed differences in indi-vidual flare source spectra and their evolution to be studiedin considerable detail.

Imaged spectra and the relative timing of sources in three flares, including the limb flareSOL2002-02-20T11:07, were studied by Krucker & Lin (2002).Sui et al. (2002) analyzedand modeled the two footpoint sources and a high, above-the-looptop hard X-ray source ob-served in this flare. Emslie et al. (2003) analyzed SOL2002-07-23T00:35 (X4.8) flare withfour HXR sources observed byRHESSI. They found a coronal source with a strong thermalcomponent, but the non-thermal component could not be studied due to severe pulse pile-up.


Battaglia & Benz (2006) studied five M-class events. Due to the smaller pile-up amount inthose events, studying the non-thermal coronal emission was possible. The results of thesestudies are summarized below.

Fig. 10.1 Top leftComposite CLEAN image of aRHESSIevent with three hard X-ray sources. The footpoints(labeled 1 & 2) are visible on the solar disc in an image made at34–38 keV. The position of the coronal source(labeled 3) high above the limb is indicated by the 50 and 80% white contours taken from a 10–12 keV image.Plots 1-3show spectra and normalized residuals over the fitted energyrange for the north footpoint (1), southfootpoint (2), and coronal source (3) (after Battaglia & Benz 2006).

10.2 Relation between coronal and footpoint sources

The quantitative relations between the footpoints and the coronal source and between thetwo footpoints can give information about the physical mechanisms at work in a solar flare.Simple models envision a beam of accelerated electrons encountering a low-density regionin the corona, leading to thin-target bremsstrahlung. Whenthe same electron beam reachesthe chromosphere, the particles are fully stopped in the dense material, producing thick-target emission. Assuming an electron power-law distribution for the electron energyE ofthe form

F (E) = AE−δ (10.1)

46 Holman et al.

producing thin-target bremsstrahlung in the coronal source, the observed photon spectrumhas spectral indexγthin = δ +1 (Equation 2.10). Reaching the chromosphere, the acceleratedelectrons will be fully stopped, producing thick-target bremsstrahlung with a photon spectralindex γthick = δ −1 (Equation 2.12). In such a simple scenario one would therefore expecta difference in the photon spectral indexγthin − γthick = 2 between the coronal source andthe footpoints. Further, the two footpoints should be of equal hardness and intensity if oneassumes a symmetric loop and symmetric injection of particles into the legs of the loop.

10.2.1 Observed difference between coronal and footpoint spectral indices

A sample of flares observed withYohkohto have coronal HXR sources was studied byPetrosian et al. (2002). They found that the spectral index of the coronal sources was, onthe average, steeper by 1 than the spectral indices of the footpoint sources. Sui et al. (2002)also found a spectral index difference of 1 for SOL2002-02-20T11:07 (C7.5) observed withRHESSI.

Battaglia & Benz (2006) found that the coronal source was softer than both footpointsfor all of their five events in nearly all analyzed time bins. Figure 10.1 (top left) shows animage of SOL2005-07-13T14:49 (M5.0) in the 34-38 keV energyband. The two footpointsare visible, as well as the 50 and 80% contours of the coronal source taken from a 10–12 keV image. Spectra and spectral fits are shown for the two footpoints and the coronalsource. The steepness of the coronal source spectrum (number 3 in the figure) relative tothe spectra from the footpoints is apparent. However, the quantitative difference betweenthe values of the spectral index obtained for the coronal source and the footpoints oftendiffered significantly from 2. For the five flares analyzed, the smallest mean difference inthe spectral indices, averaged over time, was 0.59±0.24. The maximum mean difference,averaged over time, was 3.68±0.14. These clearly contradict the theoretical expectationsummarized above. Simple thin-thick target scenarios do not seem to work in most casesand additional effects need to be considered.

Evidence for two populations of coronal source non-thermalspectra was found by Shao & Huang(2009a). They compare coronal and footpoint spectral indices at 28 hard X-ray peaks from13 single-loop flares observed byRHESSI. The spectral index in the coronal sources wasdetermined from an isothermal plus power-law fit below 30 keV, while the footpoint spec-tral indices were determined from a power-law fit at 30-60 keVphoton energies. They arguethat the coronal spectra can be divided into two groups. One,for which the coronal spectralindex is greater than 5, is well correlated with the footpoint spectral index, and the differencein the indices ranges from 2-4. For the other, where the spectral indices are anticorrelated,the coronal spectral index is less than 5, and the differencein the indices ranges from 0-2.For the group of anticorrelated spectral indices, the coronal spectral index is correlated withthe photon flux, while the footpoint spectral index is anticorrelated with the photon flux forboth groups. These are intriguing results if confirmed by future studies.

10.2.2 Differences between footpoints

No significant difference was found in the spectral indices for the two footpoints in SOL2002-02-20T11:49 (C7.5) by Krucker & Lin (2002) and Sui et al. (2002). Piana et al. (2007) in-verted count visibility spectra for this flare to obtain meanelectron flux distributions forthe footpoints. They found the mean electron flux distribution function at the northern foot-point to be somewhat steeper (∆δ ≈ 0.8) than that derived for the southern footpoint. Theyalso found the distribution function for the region betweenthe footpoints (not the coronal


source studied by Sui et al.) to be steeper than the footpointdistribution functions (∆δ ≈ 1.6relative to the southern footpoint) and to substantially steepen at energies above∼60 keV.

Krucker & Lin (2002) found that, when a connection between footpoints could be deter-mined, the footpoints brightened simultaneously (to within the∼1 s time resolution of theobservations) and had similar spectra.

Differences of 0.3 – 0.4 between the spectral indices of two footpoints in SOL2002-07-23T00:35 (X4.8) were reported by Emslie et al. (2003).

For the flares analyzed by Battaglia & Benz (2006), a significant difference was foundin only one out of five events. For all other events, the mean difference inγfp was zero withinthe statistical uncertainty.

Different spectra at the two footpoints imply an asymmetricloop. Such an asymmetrycan result, for example, from different column densities ordifferent beam fluxes and cor-responding return current energy losses in the legs of the loop. It could also result fromasymmetric magnetic trapping within the loop (e.g., Alexander & Metcalf 2002). In a studyof 53 flares showing two HXR footpoints, Saint-Hilaire et al.(2008) found that footpointasymmetry was greatest at the time of peak HXR flux and the difference in the footpointspectral indices∆γ rarely exceeded 0.6. In most cases they found the footpoint asymmetryto be inconsistent with different column densities in the two legs of the loops.

In SOL2003-10-29T20:49 (X10.0) Liu et al. (2009a) found that the brighter HXR foot-point was marginally, but consistently harder than the dimmer footpoint by∆γ = 0.15±0.13. They concluded that neither asymmetric magnetic mirroring nor asymmetric columndensityalonecan explain the full time evolution of the footpoint HXR fluxes and spectralindices. However, a self-consistent explanation might be obtained by considering these twoeffects together and/or in combination with one or more additional transport effects, such asnonuniform target ionization, relativistic beaming, and return current losses.

10.3 Spectral evolution in coronal sources

Previous observations of SHS spectral evolution (see Section 9.1) were made with full-Sunspectra which, except for over-the-limb events, are typically dominated by footpoint emis-sion. Battaglia & Benz (2006), in their imaging spectroscopy study, found that the coronalsource itself shows SHS evolution. This is illustrated in Figure 10.2. This finding impliesthat SHS is not caused by transport effects within the flare loop, but is rather a propertyof the acceleration mechanism itself. Indeed, Grigis & Benz(2006) showed that SHS canbe reproduced for electron spectra in a transit-time-damping, stochastic-acceleration model(Section 9.2).

10.4 Interpretation of the connection between footpoints and the coronal source

In the above account, emphasis was given to the difference inthe spectral index between thecoronal source and footpoints. Assuming a thin target in thecorona and a thick target at thefootpoints, one would expect a difference of two. However, whether the coronal source actsas thin- or thick-target depends on the energy of the accelerated electrons and the columndensity in the corona. Veronig & Brown (2004), for example, found coronal sources withcolumn densities high enough to act as thick targets for electrons with energies up to 60 keV.

As early as 1976, Melrose & Brown (1976) showed that magnetictrapping with colli-sional scattering of electrons out of the trap can lead to a thick-target coronal source. The

48 Holman et al.

γ

Fig. 10.2 Top: GOES1–8A light curve of SOL2003-10-24T02:54 (M7.6).Middle: RHESSI25–50 and 50–100 keV light curves near the peak of theGOESflare.Bottom: time evolution of fitted coronal source fluxat 35 keV (F35, * symbols, left log scale) and spectral index (γ , + symbols, right log scale) displaying SHSevolution (after Battaglia & Benz 2006).

coronal source transitions through a thin-thick period, with the time scale for this transitiondepending on the electron energy and the plasma density in the trap. The trapping essentiallyincreases the effective column density in the corona. Metcalf & Alexander (1999) analyzedsix flares with coronal sources observed byYohkohand found that three of the six flaresshowed properties consistent with trapping.

A simple 1-D model that described the coronal emission as intermediate thin-thick, de-pending on electron energy, was developed by Wheatland & Melrose (1995). In this modela high-density region (&1012 cm−3) is hypothesized to be present at or above the top of theflare loop. The model makes predictions for the shape of the coronal and footpoint spec-tra and the relations between them. Fletcher (1995) obtained Monte Carlo solutions to theFokker-Planck equation to show that, with the inclusion of high electron pitch angles andcollisional scattering, a compact coronal X-ray source is produced at the top of a loop with aconstant coronal density∼3×1010 cm−3. Holman (1996) showed that, even in the simple 1-


D model, a compact coronal source is produced when electronsare injected into a loop witha constant coronal density∼2×1011 cm−3 (seehesperia.gsfc.nasa.gov/sftheory/loop.htm).A compact coronal HXR source can also be produced if there is acompact magnetic trap ator above the top of the loop. Fletcher & Martens (1998) showedthat, with such a trap, a sig-nificant coronal X-ray source can be produced at plasma densities as low as∼4×109 cm−3.Petrosian & Donaghy (1999) showed that the coronal HXR source can be a consequenceof acceleration and trapping by turbulence or plasma waves.In their stochastic accelera-tion model, the difference between the coronal and footpoint spectra is explained by theenergy-dependent time scale for electrons to escape the acceleration region.

εc = εintersect

Fig. 10.3 Left: spectra for coronal source (red) and footpoints (blue) according to the model ofWheatland & Melrose (1995). The spatially integrated spectrum is shown in violet.Right: observedRHESSIspectra for the event SOL2003-10-24T02:54 (M7.6). Isothermal and power-law fits to the coronal (crosses)and footpoint (dots) spectra are shown. The vertical line indicates the predicted critical energy for the transi-tion between thin and thick target (after Battaglia & Benz 2007).

The left panel of Figure 10.3 illustrates the model of Wheatland & Melrose (1995). Thespatially integrated spectrum (violet) is the power-law spectrum (thick-target,γthick = δ −1)expected for a single-power-law electron distribution with no low- or high-energy cutoffsand no thermal component. Forε ≪ εc =

√2KN (see Equation 2.4), the spectrum is dom-

inated by thick-target radiation from the coronal source (red). There is a low-energy cutoffin the electron distribution at the footpoints atE ≈

√2KN because of the energy losses in

the coronal source. The spectrum is dominated by thick-target radiation from the footpoints(blue) whereε ≫ εc. It is in this regime that the radiation from the coronal source is thin-target and the spectral index of the coronal source is steeper by 2 than that of the footpoints.These spectra are characteristic of all the models reviewedabove.

Sui et al. (2002) compared theRHESSIobservations of SOL2002-02-20T11:07 (C7.5)to a model with a constant-coronal-density loop and no magnetic trapping. They used a finitedifference method (e.g., McTiernan & Petrosian 1990; Holman et al. 2002) to obtain steady-state solutions to the Fokker-Planck equation with collisional scattering and energy losses.Model images were convolved with theRHESSIresponse to produce simulatedRHESSIobservations for direct comparison with the SOL2002-02-20T11:07 flare images and imagedspectra. They found that, after obtaining a power-law modelspectrum with an index ofγ = 3that agreed with the observed footpoint spectra, the effective spectral index of the coronal

50 Holman et al.

source from the model (γ = 4.7) was significantly steeper than that obtained for the flare(γ = 4).

Battaglia & Benz (2007) compared the model of Wheatland & Melrose (1995) to theresults of their study of five flares observed byRHESSI. The right panel of Figure 10.3shows observed spectra and spectral fits for one particular event. The observed spectra weredominated by thermal coronal emission at low energies. Therefore, not all of the modelpredictions could be tested. However, the observed relations between the spectra did notagree with the predictions of the model. For the flare in Figure 10.3, for example, the differ-ence between the coronal source and footpoint spectral indices at the higher photon energiesis 3.8± 0.1, not 2. Also, an estimate of the column density in the coronal source gives√

2KN ≈ 10-15 keV, while the intersection of the coronal and footpoint spectra is foundto be atε ≈ 23 keV. Battaglia & Benz (2008) have found that this large difference in thespectral indices is consistent with spectral hardening caused by return current losses (seeSection 5).

11 Identification of electron acceleration sites from radioobservations

While energetic electrons excite hard X-ray emission during their precipitation into thedense layers of the solar atmosphere, they can also excite decimeter and meter wave ra-dio emission during propagation and trapping in magnetic field structures in the dilute solarcorona. The radio emission pattern in dynamic spectrogramscan give information about theelectron acceleration process, the locations of injectionof electrons in the corona, and theproperties of the coronal magnetoplasma structures.

Fig. 11.1 SOL2003-10-28T11:10 (X17.2).Left, bottom:(see Warmuth et al. 2007) 200-400 MHz radio spec-trum (Astrophysical Institute Potsdam) showing the signature of the outflow termination shock (TS, starting at11:02:47 UT).Left, top: INTEGRALcount rates at 150 keV and 7.5–10 MeV.Right:(after Aurass et al. 2007):radio source positions (Nancay Radio Heliograph, 327 MHz)overlaid on aSOHO-EIT image (11:47 UT195A). The bright areas are EUV flare ribbons in AR10486.RHESSIHXR centroids are shown as “+”. Theintegration time intervals are: for the TS source SW of AR10486 11:02:45–11:03:15 UT, for the continuumsource CONT N of AR10486 11:13–11:17 UT, respectively (see also Figure 11.2). The radio contours are at50, 70, and 99.5% of the peak flux value.

Here we take as an example SOL2003-10-28T11:10 (X17.2) (shown in Figure 11.1).Different acceleration sites can be discriminated during the impulsive and the gradual flarephases. Radio spectral data from the Astrophysical Institute Potsdam (AIP; Mann et al.


1992), imaging data from the Nancay Radio Heliograph (NRH,Kerdraon & Delouis 1997),and hard X-ray (RHESSI, INTEGRAL) data were combined in the analysis of this event.The conclusion was reached that a nondrifting, high-frequency type II radio burst signa-ture in the radio spectrum coincided with a powerful electron acceleration stage. Simul-taneously with the nondrifting type II signature, highly relativistic (≥10 MeV) electronswere observed in the impulsive phase of the flare (Figure 11.1, upper left). The radio spec-trum suggests that this can be due to acceleration at the reconnection outflow termina-tion shock (Aurass & Mann 2004), as predicted for a classicaltwo-ribbon flare (Forbes1986, Tsuneta & Naito 1998, Aurass et al. 2002). The radio source site is observed about210 Mm to the SW of the flaring active region (TS in Figure 11.1,right). In this direc-tion, TRACEandSOHO/LASCO1 C2 images reveal dynamically evolving magnetoplasmastructures in an erupting arcade (Aurass et al. 2006). For realistic parameters derived fromthese observations (the geometry, density, temperature, and low magnetic field values of∼5 Gauss), Mann et al. (2006) have found that a fully relativistic treatment of accelerationat the fast-mode outflow shock can explain the observed fluxesof energetic particles (seeZharkova et al. 2011).

Fig. 11.2 Timing of the source CONT in Figure 11.1: the NRH 327 MHz flux curve (asterisks) versus theGOES0.5-4 A flux curve (solid line, partly off-scale). Inset:SOHO/EIT image showing the radio sourcecentroid (white asterisk) andRHESSIHXR centroids as in Figure 11.1. Thick bar: the start time of GeV-energy proton injection in space (after Aurass et al. 2007).

In the main flare phase of the same event, an additional radio source (CONT in Fig-ure 11.1) was found, lasting for∼10 min, indicating the presence of another accelerationsite. No X-ray, EUV, or Hα emission was observed at the location of this radio source. Fig-ure 11.2 gives the timing and the source position with respect to the flaring active region.CONT is a m-dm-continuum source with fiber burst fine structure. Fiber bursts are excitedby whistler waves propagating along field lines of the coronal magnetic field. As marked bya bold bar in the Figure, the time of the CONT emission is also the start time of GeV protoninjection in space. Aurass et al. (2006) have shown that thissource site is not far from anopen field (particle escape) region in the potential coronalmagnetic field. The source briefly

1 Large Angle and Spectrometric Coronagraph

52 Holman et al.

flashes up already in the early impulsive phase. Based on a newmethod of fiber burst analy-sis (Aurass et al. 2005; Rausche et al. 2007), Aurass et al. (2007) argue that this source mostlikely indicates acceleration at a contact between separatrix surfaces of different magneticflux systems.

Radio observations of flares and their implications are further addressed in White et al.(2011).

12 Discussion and Conclusions

12.1 Implications of X-ray observations for the collisional thick-target model

As discussed in Section 2, the core assumption of the collisional thick-target model is thatthe spatially integrated hard X-ray emission from non-thermal electrons is bremsstrahlung(free-free radiation) from electrons that lose all their suprathermal energy through colli-sional losses in the ambient plasma as they simultaneously radiate the hard X-rays. “Simul-taneously” means within the observational integration time. This implies that all electronsthat contribute significantly to the observed radiation reach a plasma dense enough or, moreprecisely, traverse a high enough column density for all of their suprathermal energy abovethe observed photon energies to be collisionally lost to theambient plasma within the inte-gration time. For typical&1 s integration times, these conditions are met when the electronsstream downward from the corona into the increasingly denseplasma of the solar transitionregion and chromosphere.

Since the thick-target model is often implicit in our interpretation of the hard X-rayemission from flares, it is important to keep the underlying assumptions in mind and test themodel while at the same time applying it to flare observations. We have discussed above sev-eral physical processes that, if significant, change the conclusions of the simple collisionalthick-target model regarding the electron distribution produced in the acceleration region.These processes occur in either the thick-target region itself, or during the propagation of theelectrons from the acceleration region to the thick-targetregion. Only with the high spec-tral resolution and imaging ofRHESSIhas it become possible to observationally addressthese processes. Even with theRHESSIobservations, however, it is difficult to conclusivelydetermine the importance of each process.

A physical process that distorts the emitted X-ray spectrumis albedo (Section 3.4 andKontar et al. 2011). Fortunately, the albedo contribution to the X-ray spectrum can be easilycorrected on the assumption that the X-ray photons are isotropically emitted. This correctionis available in theRHESSIspectral analysis software. If the photons are significantly beameddownward, however, the distortion of the spectrum can be substantially greater than thatfrom isotropically emitted photons. An anisotropic photondistribution results from emit-ting electrons with an anisotropic pitch-angle distribution. The degree of anisotropy of theelectron pitch-angle distribution also quantitatively affects conclusions from the thick-targetmodel concerning the acceleration process. Therefore, it is important to better determine thepitch-angle distribution of the emitting electrons and thecontribution of albedo to the hardX-ray spectrum (see Kontar et al. 2011).

The simple collisional thick-target model assumes that thetarget plasma is fully ionized.We have seen, however, that a nonuniformly ionized target region can produce an upwardkink, or “chicane,” in an otherwise power-law X-ray spectrum (Section 4). This spectralshift can provide a valuable diagnostic of the ionization state of the target plasma and itsevolution. It is likely, however, that the power-law spectrum below the chicane is hidden by


thermal radiation. The chicane is then observed only as a downward break in the spectrum atenergies above those dominated by the thermal emission. Theupper limit on the magnitudeof the break provides a method for ruling out nonuniform ionization as the sole cause of largespectral breaks. To further distinguish this break from spectral breaks with other causes, itis important to better determine the degree of ionization asa function of column density atthe thick-target footpoints.

Return-current energy losses can also produce a downward break in the X-ray spectrum(Section 5). The break energy depends on both the thermal structure of the plasma in theflare loop and on the non-thermal electron flux density distribution. These spectral modifi-cations and their evolution throughout flares provide an important test for the presence ofinitially un-neutralized electron beams and the return currents they must drive to neutralizethem. AlthoughRHESSIobservations provide substantial information about the structureand evolution of flare spectra, only a lower limit on the electron flux density can usuallybe determined. Observations and analysis sufficiently accurate and comprehensive to verifythe presence of return current energy losses as the cause of aspectral break are yet to beobtained. On the other hand, significant evidence exists (Section 5.3) indicating that returncurrent losses do have an impact on flare hard X-ray emission.

A thorough comparison of measured flare spectra with theoretical spectra computedfrom models incorporating collisional and return current energy losses (including their ef-fect on the angular distribution of the non-thermal electrons), as well as nonuniform targetionization and albedo, is still needed. Spectral fitting alone, however, is not likely to distin-guish the importance of these different mechanisms. Comparison of the time evolution ofthe spectra, as well as of the spatial structure of the X-ray emission, with expectations wouldcertainly enhance the possibility of success for such an endeavor.

The analysis of the evolution of X-ray source positions and sizes with photon energyand time provides another important test of the collisionalthick-target model (Section 7).For these flares that show non-thermal source evolution in the corona and upper transitionregion, the source position and size are sensitive to the energy losses experienced by thenon-thermal electrons. They are, in fact, sensitive to the very assumption that the sources areproduced by electrons as they stream downward from an acceleration region higher in thecorona. Further studies of the evolution of these coronal X-ray sources should substantiallyclarify the applicability of the collisional thick-targetmodel.

For completeness, we note that under some circumstances other radiation mechanismsmay significantly contribute to the X-ray emission from non-thermal electrons. The possibil-ity that recombination (free-bound) radiation from the non-thermal electrons is sometimesimportant is discussed in Brown & Mallik (2008, 2009) (also see Kontar et al. 2011). How-ever, the contribution of non-thermal free-bound radiation has recently been found to be lesssignificant than originally estimated (Brown et al. 2010). MacKinnon & Mallik (2010) haveconcluded that inverse Compton radiation may significantlycontribute to the X-ray/γ-rayemission from low-density coronal sources.

Another testable aspect of the collisional thick-target model is the heating of the flareplasma by the non-thermal electrons. If the flare plasma is primarily heated by these elec-trons and the thick-target region is primarily in the chromosphere and lower transition re-gion, heating originating in the footpoints and expanding into the rest of the flare loopthrough “chromospheric evaporation” should be observed. On the other hand, if the loop isdense enough for the thick-target region to extend into the corona or if return-current heatingis important, localized coronal heating and different ion abundances should be observed.

It has generally been difficult to establish a clear connection between the location andevolution of X-ray sources produced by non-thermal electrons and by thermal plasma at

54 Holman et al.

different temperatures. This is largely because of a lack ofhigh-cadence EUV images cov-ering a broad range of coronal and transition region temperatures prior to the launch ofSDO. Future studies of the coevolution of non-thermal X-ray sources and thermal sources inflares will be important in determining the extent to which heating mechanisms other thancollisional heating by non-thermal electrons is significant.

Predicting the expected evolution of the heated plasma is hampered by insufficientknowledge of the dominant heat transport mechanisms. We have seen evidence that manyflares cool by classical thermal conduction or radiation once the heating has subsided (Sec-tion 8.4), but this is not likely to be the dominant transportmechanism during rapid heating.Nevertheless, the spatial evolution of flare X-ray sources has so far been found to be con-sistent with chromospheric evaporation (Section 7.2). Also, the Neupert effect, observedin most flares, and Doppler-shift measurements qualitatively support the thick-target model(Section 8.3), but these do not rule out the possibility of other heating mechanisms tem-porally correlated with the electron beam collisional heating. As discussed in Section 3,substantial progress has been made in deducing the energy flux (total power) carried bynon-thermal electrons, but we usually can deduce only a lower limit to this energy flux. Con-tinuing studies of flares similar to SOL2002-04-15T03:55 (M1.2) and the initially cooler,early-impulsive flares (Section 3.5) may provide a better handle on this energy flux for com-parison with thermal evolution. The thermal properties, energetics, and evolution of flaresare discussed further in Fletcher et al. (2011).

12.2 Implications of X-ray observations for electron acceleration mechanisms and flaremodels

In Section 10 we addressed the X-ray spectra of hard X-ray sources sometimes observedabove the top of the hot loops or arcades of loops observed in flares. We reviewed resultsindicating that the spectra are qualitatively, but not quantitatively consistent with expecta-tions for electrons passing through a thin-target or quasi-thick-target region on their way tothe thick-target footpoints of the flare loops. The apparentfailure of these relatively simplemodels is probably a manifestation of the more complex above-the-looptop X-ray sourcestructure revealed byRHESSIobservations.

BeforeRHESSI, time-of-flight delays in hard X-ray timing indicated that electrons wereaccelerated in a region somewhat above the looptops of the hot flare loops in most flares(Section 8.1). Also, cusps were observed at the top of flare loops byYohkoh(e.g., Sec-tion 10), indicating a magnetic connection to the region above the hot loops.

RHESSIimages have revealed flares with double coronal sources, oneat or just abovethe top of the hot loops and the other at a higher altitude above the lower source. Thecentroid of the lower source is higher in altitude at higher X-ray energies, while the cen-troid of the upper source is lower in altitude at higher X-rayenergies, indicating that en-ergy release occurred between these coronal sources (Sui & Holman 2003; Sui et al. 2004;Liu et al. 2008, 2009b). In one flare, the upper source accelerated outward to the speed of asubsequent coronal mass ejection. The white-light coronagraph on theSolar Maximum Mis-sion(Webb et al. 2003), the Large Angle and Spectrometric Coronagraph (LASCO) and theUltraviolet Coronagraph Spectrometer (UVCS) onSOHO(Ko et al. 2003; Lin et al. 2005),RHESSI(Sui et al. 2005b), andTRACE(Sui et al. 2006a) have all provided direct evidencefor the presence of an extended, vertical current sheet above the hot flare loops and belowthe coronal mass ejection associated with eruptive flares. These and related observations arediscussed further in Fletcher et al. (2011).


These recent observations strongly support the “standard”model of eruptive solar flares,in which the hot flare loops build up below a vertical current sheet where inflowing magneticfields reconnect and a magnetic flux rope forms above the current sheet to become a coronalmass ejection (see Fletcher et al. 2011; Zharkova et al. 2011). The rate of electron acceler-ation has been observed to be correlated with the rate at which magnetic flux is swept upby the expanding footpoints of flare loops and with the rate oflooptop expansion (Qiu et al.2004; Sui et al. 2004; Holman 2005), indicating that the electron acceleration rate is cor-related with the rate of magnetic reconnection. On the otherhand, the observations alsoindicate that the rate of electron acceleration in the impulsive phase of flares is greatestbeforea large-scale current sheet or soft X-ray cusp is observed (Sui et al. 2008).

Initially, when the electron acceleration rate is highest,the current sheet may be shortand associated with slow-mode shock waves, as in Petschek reconnection. Fast reconnectionjets (e.g., Wang et al. 2007) can stream upward and downward from the current sheet, possi-bly ending in fast-mode shock waves where they collide with slower magnetized plasma atthe flare loop tops and the lower boundary of the magnetic flux rope (termination shocks).The pair of above-the-looptop X-ray sources may be associated with these fast-mode shockwaves. We have described possible evidence for these shock waves from radio observationsin Section 11.

The most difficult task is determining the dominant acceleration mechanism or mech-anisms responsible for the energetic particles. The regionabove the flare loops contains orcan contain quasi-DC electric fields, plasma turbulence, slow- and fast-mode shock waves,and collapsing magnetic traps, allowing for almost any acceleration mechanism imaginable.The problem is as much one of ruling out mechanisms as of finding mechanisms that work(cf. Miller et al. 1997). Acceleration mechanisms are addressed in Zharkova et al. (2011).

In Section 9 we addressed the soft-hard-soft evolution of flare X-ray spectra. This spec-tral evolution could occur during the propagation of the electrons from the accelerationregion to the thick-target footpoints. Return current losses, with their dependence on theelectron beam flux (Section 5), for example, could be responsible for this evolution. How-ever, the observation that above-the-looptop sources alsoshow this spectral evolution (Sec-tion 10.3) indicates that it is a property of the acceleration process rather than electron beampropagation. We saw in Section 9.2 that the soft-hard-soft behavior can be reproduced inthe acceleration region if the acceleration or trapping efficiency first increases and then de-creases.

Flares displaying soft-hard-harder spectral evolution are of special interest, because theyhave been shown to be associated with high-energy proton events in space (Kiplinger 1995;Saldanha et al. 2008; Grayson et al. 2009). What is the connection between the appearanceof energetic protons in space and X-ray spectral hardening late in flares? The answer to thisquestion is important to both space weather prediction and understanding particle accelera-tion in flares.

12.3 Implications of current results for future flare studies in hard X-rays

What characteristics should a next-generation hard X-ray telescope have to make substantialprogress in understanding electron propagation and acceleration in flares? The advancesmade withRHESSIhave depended on its high-resolution count spectra that could generallybe convolved with the detector response to obtain reliable photon flux spectra. These havebeen the first observations to allow detailed information about the evolution of acceleratedelectrons and associated hot flare plasma to be deduced for many flares. Equally important

56 Holman et al.

has been the ability to produce hard X-ray images in energy bands determined by the userduring the data analysis process. This imaging capability has been critical to determining theorigin of the X-ray emission at a given photon energy and in obtaining spectra for individualimaged source regions. These high-resolution imaging spectroscopy capabilities will remainimportant for continued progress.

RHESSI’s X-ray imaging capability has allowed a clear spatial separation to be made formany flares between footpoint sources with non-thermal spectra at higher energies and loop-top sources with thermal spectra at lower energies. However, in the energy range of overlapbetween∼10 keV and 50 keV, where both types of sources may coexist, it is often difficultto distinguish weaker coronal sources (both thermal and non-thermal) in the presence of thestronger footpoint sources. This is because of the limited dynamic range of<100:1 (and sig-nificantly less for weaker events) that is possible in any oneimage made fromRHESSIdata.This is a consequence of the particular form of the Fourier-transform imaging technique thatis used. Thus, in most flares the usually intense footpoints mask the much weaker coronalhard X-ray sources that can sometimes be seen in over-the-limb flares when the footpointsare occulted (e.g., Krucker & Lin 2008). In fact, these coronal hard X-ray sources can ex-tend to high energies (up to∼800 keV, Krucker et al. 2008b) and seem to be non-thermal inorigin, thus making them of great interest in locating and understanding the particle acceler-ation process. It is important to study these non-thermal coronal sources in comparison withthe footpoint sources, something that is currently not possible with RHESSI’s limited dy-namic range except in the few cases with exceptionally strong coronal emission (see Section10). In addition, again because of theRHESSIdynamic range and sensitivity limitations, ithas not generally been possible to observe the thin-target bremsstrahlung emission from thecorona that must be present from the electrons streaming down the legs of magnetic loopsand also from electrons streaming out from the Sun and producing type-III bursts (see, how-ever, Krucker et al. 2008c; Saint-Hilaire et al. 2009). For all of these reasons, a significantlygreater dynamic range will be an important goal for future advanced solar hard X-ray in-struments.

Flares at the solar limb for which the hard X-ray footpoints are occulted by the diskprovide an important way of observing coronal hard X-ray sources, but these flares do notallow a comparison to be made between the coronal emission and the thick-target foot-point emission. A possible substitute for a high-dynamic-range instrument is hard X-rayobservations from two or more spacecraft. Under the right conditions, one spacecraft canobserve all the flare emission while the other observes only the coronal emission, with thefootpoint emission occulted by the solar disk. Multi-spacecraft observations would also beimportant for deducing the directivity of the flare emission(especially in conjunction withX-ray polarization measurements – see Kontar et al. 2011) and 3-D source structure. Thismulti-spacecraft approach, however, limits the number of flares for which the coronal andfootpoint emissions can be compared.

Hard X-ray timing studies have provided valuable information about electron propa-gation and the location of the acceleration region (Section8). Since the time of flight ofenergetic electrons from a coronal acceleration region to the thick-target loop footpoint istypically ∼10–100 milliseconds, the photon count rate must be high enough to distinguishdifferences in flux on these time scales. Time-of-flight studies have not been successful withRHESSI, because of its relatively low collecting area and, therefore, count rate. An instru-ment with the collecting area and pulse-pileup avoidance ofCGRO/BATSE, and the imagingand spectral resolution ofRHESSI, would provide a new generation of studies on the char-acteristic time scales of propagation for the hard-X-ray-emitting electrons accelerated inflares. Alternatively, smaller instruments sent closer to the Sun on, for example,Solar Or-


biter or Solar Probecould achieve the required sensitivity. Flare studies on these time scaleswould provide important insights into the physical processes that impact the accelerationand propagation of energetic electrons in flares.

Acknowledgements We thank the chapter editor, Brian Dennis, and the two reviewers for comments thatled to many improvements to the text. GDH acknowledges support from theRHESSIProject and NASA’sHeliophysics Guest Investigator Program. MJA acknowledges support from NASA contract NAS5-98033 oftheRHESSImission through University of California, Berkeley (subcontract SA2241-26308PG), and NASAcontract NAS5-38099 for theTRACEmission. HA acknowledges support by the German Space AgencyDeutsches Zentrum fur Luft- und Raumfahrt(DLR), under grant No. 50 QL 0001. MB acknowledges supportby the Leverhulme Trust. PCG acknowledges support from NASAcontract NNM07AB07C. EPK acknowl-edges support from a Science and Technology Facilities Council Advanced Fellowship. NASA’s AstrophysicsData System Bibliographic Services have been an invaluabletool in the writing of this article.

58 Holman et al.

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Index

above-the-looptop sources, 55abundances

bremsstrahlung efficiency, 7evaporation, 53

accelerationand large-scale processes, 18DC electric field, 15, 44from thermal plasma, 10list of mechanisms, 55shock, 51stochastic, 40, 43, 49transit-time damping, 40two-stage, 33, 43

acceleration region, 33, 40, 50, 52, 54and SHH pattern, 44and SHS pattern, 55distinguished from energy-loss region,

3, 52escape from, 25, 49extended, 32height, 4, 53return current, 4separatrix structure, 52spectral cutoff, 15

albedo, 17, 26, 38, 52and low-energy cutoff, 14dip in spectrum, 14low-energy cutoff, 17

Ampere’s Law, 22atmospheric models

“spicular extended chromosphere”,27

FAL, 27hydrostatic, 27semi-empirical, 27VAL, 27

beamsinduced fields, 22propagation, 18super-Alfvenic, 17un-neutralized, 53

bremsstrahlung, 3Bethe-Heitler cross-section, 7–9cross-section, 5

Kramers approximation, 7, 8NRBH, 7software, 9thick-target, 46thin-target, 46

caveatslow-energy cutoff, 14

chicane, 52chromospheric density model, 27chromospheric evaporation, 22, 26, 29, 30,

32, 34, 53, 54collisions

energy losses, 18particle energy losses, 3–5

column density, 6, 13, 18, 27, 47, 52Compton Gamma Ray Observatory (CGRO),

15, 33, 56Compton Gamma-Ray Observatory., 15conduction fronts, 26convective-diffusive equation, 41cooling

conductive, 33, 34, 36, 37delay, 37

radiative, 33, 34coronal mass ejections (CMEs), 17, 54coronal sources, 44Coulomb logarithm

electron-electron collisions, 5electron-hydrogen collisions, 18

cross-sectionsBethe-Heitler, 7Kramers approximation, 7, 8

illustration, 8NRBH, 7, 8

comparison with Kramers, 8comparison with relativistic, 9illustration, 8

current sheets, 54

differential emission measure, 34double layer, 26downflows

soft X-rays, 31

electric fields, 15

64


and beam deceleration, 23and soft-hard-soft pattern, 43DC, 44double layers, 26return current, 22self-field, 4, 22self-induced, 22

electron beamsand induced fields, 22

electronsaccelerated

energy content, 3acceleration from thermal plasma, 40collision losses, 12distribution function

and turbulence, 25beam, 4density, 9, 22, 41flux, 7flux density, 5high-energy cutoff, 9, 44low-energy cutoff, 3, 9, 10, 24,

34mean electron flux, 6, 7, 13, 17,

25, 46total flux, 10total power, 10, 15, 54

energy flux density, 22, 24escape, 40

energy-dependent, 49escape time, 41high-energy cutoff, 9mean electron flux, 6pitch-angle distribution, 25relativistic, 51super-Alfvenic beams, 17time-of-flight (TOF) analysis, 33transport, 40, 44

emission measuredifferential, 34

model, 36eras

pre-RHESSI, 38

filling factorhard X-rays, 24

flare (individual)SOL1980-06-27T16:17 (M6.7)

low-energy cutoff, 15

SOL1992-01-13T17:25 (M2.0)above-the-looptop source, 4

SOL2002-02-20T11:07 (C7.5)chromospheric density structure,

27, 28coronal hard X-ray source, 46coronal source, 44coronal source vs. footpoint spec-

tra, 49nonuniform ionization, 20

SOL2002-02-20T11:49 (C7.5)footpoint differences, 46

SOL2002-02-26T10:27 (C9.6)multi-thermal time delays, 36

SOL2002-03-17T19:31 (M4.0)nonuniform ionization, 20

SOL2002-04-15T03:55 (M1.2)electron energy flux, 54low-energy cutoff, 16spectral fits, 16



SOL2002-07-23T00:35 (X4.8)coronal hard X-ray source, 44energy in non-thermal electrons,

14footpoint differences, 47nonuniform ionization, 19–21

SOL2002-08-20T08:25 (M3.4)low-energy cutoff & albedo, 14

SOL2002-11-09T13:23 (M4.9)illustration, 38spectral evolution, 38

SOL2002-11-28T04:37 (C1.0)illustration, 31X-ray source motion, 29, 31

SOL2003-10-24T02:54 (M7.6)coronal source vs. footpoint spec-

tra, 49spectral evolution, 48

SOL2003-10-28T11:10 (X17.2)illustration, 50, 51radio emission, 50

SOL2003-10-29T20:49 (X10.0)footpoint differences, 47

SOL2003-11-13T05:01 (M1.6)density profiles, 30

66 Holman et al.

illustration, 29, 30increasing density, 32Neupert Effect, 34source locations vs. photon energy,

29X-ray brightness profiles, 30X-ray source motion, 29

SOL2004-01-06T06:29 (M5.8)chromospheric density & magnetic

structure, 27SOL2004-11-03T03:35 (M1.6)

hard-soft-hard, 38SOL2005-01-19T08:22 (X1.3)

high low-energy cutoffs, 17quasi-periodic oscillations, 17soft-hard-harder, 17

SOL2005-07-13T14:49 (M5.0)coronal hard X-ray source, 45, 46

flare modelsand X-ray observations, 54multithread, 37standard, 55

flare typesearly impulsive, 17, 24, 31, 32, 54two-ribbon, 35

flaresmodel, 55

Fokker-Planck equation, 22, 48, 49footpoints

altitude of sources, 27and coronal sources, 44and loop legs, 28

illustration, 29asymmetry, 47hard X-rays, 3height structure, 28, 29simultanteity, 47spectral similarity, 47

Fraunhofer linesHα , 24Hβ , 24

free-bound emission, 53free-free emission, 3frequency

Larmor, 25proton, 41

plasma, 25FWHM, 27

GOES, 24, 31, 34, 39gradual phase, 37, 39

hard X-rays, 3above-the-looptop source, 4, 44albedo, 14, 17, 52chicane, 52coronal sources, 44, 46, 47, 49, 56

spectral evolution, 47thick-target, 47Yohkoh, 48

correlation with EUV, 35energy dispersion, 33filling factor, 24flux saturation, 24footpoint sources, 3, 7, 17, 27, 46hard-soft-hard, 38height dependence, 27, 28, 53

illustration, 30height dispersion, 4, 29inverse Compton radiation, 53loops, 28occulted, 47, 56polarization, 56relation between footpoint and coro-

nal sources, 45soft-hard-harder, 37, 39, 55soft-hard-soft, 4, 20, 37, 39, 47, 55source centroids, 28source sizes, 32spectral break, 24spectral evolution, 24, 37, 53

interpretation, 40soft-hard-harder, 17

spectral flattening, 20, 23illustration, 23

spectral index, 4, 7, 9, 39, 46spectral interpretation, 18spectral parametrization, 38thick-target, 3, 6–9, 18, 45, 52thin-target, 6–9, 45, 56time delays, 4, 32, 54, 56

Hinotori, 15

image dynamic range, 56imaging spectroscopy, 44, 56impact polarization, 24impulsive phase, 37INTEGRAL, 50


inverse Compton radiation, 53inverse problem

for X-ray spectra, 13ionization state, 3, 52

nonuniform distribution, 18particle energy loss rate, 6, 18

jetsreconnection outflow, 55

Langmuir waves, 25loops

hard X-rays, 28time-dependent structure, 29

legs, 29looptop sources, 55low-energy cutoff, 3, 9, 10, 15, 17, 24, 34

above 100 keV, 17and return current, 21and time-of-flight analysis, 17plateau, 12sharp, 10turnover, 12

magnetic fieldand return current, 22scale height, 27

magnetic structuresarcades, 35collapsing traps, 55cusps, 4, 55flux tubes, 27loss cone, 25separatrix, 52trapping, 33

magnetic trapping, 33, 38, 41–43, 47, 49magnetization, 26mean electron flux, 6, 13, 17

Neupert effect, 17, 34, 54theoretical, 34theory, 34

non-uniform ionization, 3, 18, 52

occulted sources, 28, 56Ohm’s Law, 22

plasma instabilitiesanomalous Doppler resonance, 25beam-plasma, 25

bump-on-tail, 12, 25, 26simulation, 25

electron-cyclotron maser, 25gyrosynchrotron maser, 26ion-acoustic, 26loss-cone, 25return-current, 25Weibel, 26

plasma turbulence, 25, 55polarization, 56

Hα , 24precipitation, 21proton events, 38, 39, 55

and soft-hard-harder, 39pulse pileup, 14, 56

quasi-periodic pulsationsmagnetoacoustic waves, 17

radiative cooling, 33radio emission, 4, 17, 50

fiber burst, 51gyrosynchrotron, 3microwaves, 9type II burst, 51

recombination radiation (non-thermal), 53reconnection, 17

and particle acceleration, 44and super-Alfvenic beams, 17outflow, 55Petschek, 55rate correlated with acceleration, 55termination shock, 51

resistivity, 26return current, 4, 15, 21, 22, 24, 40, 47,

53, 55and beam stability, 25and double layers, 26and impact polarization, 24and soft-hard-soft, 44and spectral lines, 24charge carrier, 26energy losses, 24, 32, 42, 50, 53

and low-energy cutoff, 26instability, 25

RHESSI, 4dynamic range, 56effective collecting area, 33imaging spectroscopy, 44, 56

68 Holman et al.

pulse pileup, 14spatial resolution, 4, 27spectral resolution, 27, 34, 55

ribbons, 35

satellitesCGRO, 15, 33, 56GOES, 24, 31, 34, 39Hinotori, 15INTEGRAL, 50RHESSI, 4SDO, 54SMM, 15, 54SOHO, 51, 54

LASCO, 51Solar Orbiter, 56Solar Probe, 56TRACE, 35, 51, 54Yohkoh, 44, 46, 48

shocksinterplanetary

electron acceleration, 2particle acceleration, 51radio emission, 55slow mode, 55termination, 51, 55

soft X-rayscooling

and time delays, 33coronal sources, 54

double, 54downward motions, 31height dependence, 29–31multi-thermal, 37multi-thermal modeling, 35

illustration, 35parallel motions, 31

illustration, 31soft-hard-soft, 20

and trapping efficiency, 42and turbulence, 43

SOHO, 51, 54solar energetic particles (SEPs)

associated with hard X-rays, 55Solar Maximum Mission, 15, 54Solar Orbiter, 56Solar Probe, 56space weather, 55spectrum

power-law, 9suprathermal populations, 52

and thick-target model, 3

termination shockillustration, 50

thermal conduction, 33, 34, 37, 54thick-target model

and spectral “dip”, 14collisional, 6description, 3filamentation, 24height dependence, 31implications, 52need for return current, 22standard, 3summary, 5variable ionization, 18

thin target, 5time-of-flight analysis, 17, 33, 54TRACE, 35, 51, 54transit-time damping, 40transition region, 18, 20, 29

non-hydrostatic, 28transport

electrons, 40trapping

in turbulence, 42magnetic, 4, 33, 55time scale, 33

turbulence, 25and electron distribution function, 25and trapping, 49MHD, 43particle acceleration, 40

visibilities, 27and electron flux maps, 28

wavesion sound, 26plasma, 25whistlers, 51

Yohkoh, 4, 44, 46, 48

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Implications of X-ray Observations for Electron Acceleration and Propagation in Solar Flares

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